Strongly zero-dimensional and ultranormal definition and basic properties

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.

[Moniker1998]

Here I introduce strongly zero-dimensional, ultranormal, and basic equivalences, namely:

ultraparacompact iff ultranormal + paracompact
ultranormal iff strongly zero-dimensional + normal

@prabau

[Moniker1998]

Sorry I forgot to revert the changes made to already existing theorems

[Moniker1998]

Now it should be ready for review and/or discussion

[prabau]

(repeating #1414 (comment))

Whatever we decide, we should have specific references using that name,
as well as mention other different uses of that name in the literature
and maybe explain why we chose the one here.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

[Moniker1998]

Whatever we decide, we should have specific references using that name,

That's there. Ultranormal is probably also in Engelking though I didn't check, I just copied references for ultraparacompact.

as well as mention other different uses of that name in the literature

Sure, that should boil down only to strongly zero-dimensional.

I recall Charalambous calls ultranormal spaces to be strongly zero-dimensional, and Munkres probably calls strongly zero-dimensional spaces to be ultraparacompact, if that definition even exists there.

And also explain what alternative names are used for the concept here, again with some references, especially if they are important references in the field.

I don't know what you mean

[Moniker1998]

@prabau here's a reference for ultranormal, and also some explanation for why we would want to use this definition of strongly zero-dimensional (that it's equivalent to $\beta X$ being zero-dimensional for Tychonoff spaces). If you want we can cite a theorem in Engelking here as well.

For ultraparacompact I guess you'd have to check what definition of strongly zero-dimensional Munkres uses, I don't use Munkres so idk what's there

[prabau]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?
(I'd say Munkres does not matter, it was a text for undergraduates and he sometimes uses his own made-up terminology.)

I did not check Nagata's book. Need to check later.

Also Encyclopedia of general topology (Hart, Nagata, Vaughan) seems to be using the same as Engelking, assuming Tychonoff.

So we should mention Engelking and that we don't assume Tychonoff here.
And mention what Charalambous uses, which is different (same as another notion we have ?)

[Moniker1998]

I would say Engelking, Charalambous and Nagata's book are kind of authoritative for dimension theory. Do you have others?

No, these are the only ones I know. There's a chapter in Gillman and Jerison that talks about dimension theory, but they don't mention strongly zero-dimensional spaces by name.

And mention what Charalambous uses, which is different (same as another notion we have ?)

I mentioned this in definition of Ultranormal.

So we should mention Engelking and that we don't assume Tychonoff here.

How would you fit it in? I don't know how to do it without making the definition awkward

[prabau]

Note: Engelking 6.2.12 mentions: for $X$ Tychonoff, $X$ is strongly zero-dim iff $\beta X$ is strongly zero-dim.
But if $\beta X$ is only zero-dim, how does one deduce strongly zero-dim?

[Moniker1998]

Note: Engelking 6.2.12 mentions: for X Tychonoff, X is strongly zero-dim iff β X is strongly zero-dim. But if β X is only zero-dim, how does one deduce strongly zero-dim?

For compact Hausdorff spaces, strongly zero-dimensional, zero-dimensional and totally disconnected are all equivalent

[Moniker1998]

I don't know why he chose such awkward theorem (probably because $\dim X = \dim \beta X$), we could cite Gillman and Jerison instead for example, theorem 16.17

[prabau]

In any case, I am not sure adding this characterization in terms of $\beta X$ is that useful here. Do you have strong opinions about that one?

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension $0$? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

[Moniker1998]

In any case, I am not sure adding this characterization in terms of β X is that useful here. Do you have strong opinions about that one?

What do you mean? You were asking for reasons to call this particular property as strongly zero-dimensional.

On the other hand, for Tychonoff spaces, isn't strongly zero-dim the same as having covering dimension 0 ? I know we don't yet have this kind of dimension theory notions in pi-base, but it seems it could useful to mention with a reference. What do you think?

That depends on what you mean by covering dimension. It could either mean strongly zero-dimensional like here, ultranormal, or ultraparacompact.

Covering dimension using cozero sets gives you strongly zero-dimensional and for this one you have $\dim X = \dim \beta X$.

Covering dimension using open sets gives you ultranormal.

Covering dimension using arbitrary size open covers gives you ultraparacompact.

[prabau]

I don't know why he chose such awkward theorem (probably because dim ⁡ X = dim ⁡ β X ), we could cite Gillman and Jerison instead for example, theorem 16.17

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

[Moniker1998]

That's a good theorem. Do Gillman & Jerison assume Tychonoff/completely regular for the whole chapter 16?

Yes but otherwise $\beta X$ doesn't make sense so we do need Tychonoff.

[prabau]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

[Moniker1998]

Due to all the variations of the different definition of covering dimension, maybe it's preferable not to mention covering dimension for now.

I believe that the dimension $\dim_0$ of Charalambous is the most general covering dimension.

In proposition 11.2 he writes $\dim_0 X = \dim X$ for normal space $X$. I believe that if $\dim'$ is the covering dimension of Munkres then we also have $\dim' X = \dim X$ for a normal paracompact space $X$.

Also I'm sure $\dim X < \infty$ implies that $X$ is normal, and I believe that $\dim' X < \infty$ should imply $X$ is normal and paracompact.

So the real difference should be only in the "domains" that those dimension functions have.

Basically the only difference between $\dim$ and $\dim_0$ is that $\dim$ is not suitable to work with non-normal spaces as it assigns infinite values for spaces we would like to have finite covering dimension. Etc.

So concluding, if we are to refer to covering dimension we should probably go with the function $\dim_0$ of Charalambous

[prabau]

Nagata does not seem to use the term "strongly zero-dim".

[prabau]

Thanks for fixing Tychonoff (I had meant P6 but wrote P7).

In the terminology paragraph, should we add:
"Two sets $A,B\subseteq X$ are separated by a clopen set$U$ if $A\subseteq U\subseteq X\setminus B$."
Or is it obvious?

[Moniker1998]

@prabau see, this is why I prefer the "disjoint clopen sets" definition because it's symmetric and this one isn't.
To me it's obvious but a lot of things that are obvious to me are not obvious to others so it's not like my opinion has heavy weight on things here

I'd wager it's obvious but then again I'd also wager that zero and cozero set definitions are should be part of every person who's interested enough into topology and shouldn't be explained. But I included them for clarity anyways.

[prabau]

Let's see other people's opinion about the previous comment and this one. @pzjp@yhx-12243: what do you think?

"separated by a clopen set" is also how the definitions of strongly zero-dimensional are given in Encyclopedia of general topology for example (p. 323). I thought that was a good way to express it, but if the majority prefers it, I am fine with the other one too.

It is true about "disjoint clopen sets" being more symmetric phrasing, superficially.
But maybe it slightly complicates the proofs and the intuition because instead of dealing with just one clopen set and its complement (which is a symmetric thing), one has to think of two semi independent clopen sets ...

Also, strongly zero-dimensional and ultranormal are kind of more advanced concepts. If anyone has progressed in topology to the point that they are trying to understand those, it seems they would have enough basic understanding to not have to chew everything for them.

I am ok with not explaining the meaning of "separated by a clopen set" if we decide to go with that.

[prabau]

Compare the proposed change above for T766. Nothing wrong with the previous, but doesn't it look slightly simpler with the "separated by a clopen set" phrasing?

[Moniker1998]

@prabau I mean the simplicity comes from us not explaining why $Z_1, Z_2$ being separated by clopen set would imply that $A_1, A_2$ are separated by clopen set so maybe it's not all that simpler

Maybe simpler in a sort of deceiving sense, or conceptually simpler. But this phrasing could be used just as well with the other definition.

[prabau]

@Moniker1998 I agree it's about the same either way.

@pzjp Can you give your opinion on #1415 (comment) and #1415 (comment) ? What do you think we should choose for the definition of "strongly zero-dimensional"?

[Moniker1998]

What do you think we should choose for the definition of "strongly zero-dimensional"?

I mean I don't think we have much of a choice here no matter what.

To be honest I'm glad that the terminology ultranormal and ultraparacompact even exists so that we don't need to have 3 different definitions of strongly zero-dimensional

[prabau]

I mean I don't think we have much of a choice here no matter what.

The concept we have chosen is the right one. Just need to decide the preferred phrasing: "separated by disjoint clopen sets" or "separated by a clopen set".

@pzjp@yhx-12243

[pzjp]

I'm positive "disjoint clopen sets" are necessary phrasing for ultranormal. But I do not have strong preference about S0dim.

Also is we allow $X$ to be non-empty right? The empty space is more likely to be a controversial case.

[prabau]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

I can fix the phrasing later, as I want to do some further minor tweaks to that README.

[Moniker1998]

Yeah, Moniker meant we consider the empty set to be strongly zero-dimensional in pi-base.
It simplifies some theorems, so we don't have to add empty/non-empty conditions in many theorems.

Yes, and it also agrees with our already existing convention for zero-dimensional spaces. They also allow empty space.

[prabau]

What do you think of this:

Any two disjoint zero-sets are separated by disjoint clopen sets. (The two clopen sets can always be chosen to be complement of each other.)

[Moniker1998]

I would change "are" for "can be".

Maybe instead of text definition we could add symbols to be more illustrative. I think that would be better if we want to do something like that

[prabau]

Any two disjoint zero-sets $Z_1$, $Z_2$ are contained in disjoint clopen sets $U_1$, $U_2$. (One can always choose $U_1$ and $U_2$ to be complement of each other.)

(reverting to your original "are contained in")

I'll make a commit later, together with a few other things, and then we can discuss further.

[prabau]

Added an update to P146 (ultraparacompact) that was necessary for the proof of T764.

[prabau]

I updated the README pages for P217 and P218.

For P217: in the section about the references Engelking thm 6.2.4 is already mentioned earlier when giving the equivalences between the definitions, so no need to repeat it. Also, indicated precisely what Engelking means by "strongly zero-dimensional" and how we differ. I put Charalambous at the end, because he does not use the same terminology. Also indicated precisely what he means by "strongly zero-dimensional".

For the formatting, I used three bullet points (including for the first item) so that all three characterizations would be at the same level. Note that usually we don't put a bullet for the first characterization. But I compared with what it would look like without the bullets, or without the first bullet; somehow it did not look as good. This should be looked at on the web page or experimented in the preview window. Please compare also with P218, which does not have bullets, but it's only two characterizations, so it seems acceptable.
Please check and see what you think. It would be good to discuss the best format.

For P218: made it more parallel to P217. Also, in the discussion of Charalambous I removed references to large inductive dimension vs. covering dimension, which does not need to be here. What is on page 9 of the book is sufficient for people to figure out, if they care about it. We don't need to explain more for now (can be modified in the future when more general dimension functions get introduced).

[Moniker1998]

I think it's fine

[prabau]

@pzjp@yhx-12243 Do you have anything else you want to comment about this PR?

[pzjp]

I have nothing to add here.