New space: Pseudocircle

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.

[yhx-12243]

No description provided.

[yhx-12243]

More references are WANTED if possible.

[prabau]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what $K$ means if we use that notation. Making a partial suggestion below.

What about something like below maybe:
For the first paragraph:
Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say:
this induces a continuous function $F \colon K \times I \to K$ such that ...
(i.e., the continuity follows from the the first paragraph).

[yhx-12243]

S51-P199: I think mentioning Whitehead's theorem (Engelking 3.3.17) would be helpful. Also, need to specify what K means if we use that notation. Making a partial suggestion below.

What about something like below maybe: For the first paragraph: Let $I=[0,1]$. We view {S51} $K$ as the quotient space of {S25} with the quotient map $p:\mathbb R\to K$ defined by $p(i)=2i$ and $p(x)=2i+1$ if $i<x<i+1,$ for all $i\in\mathbb Z$. By Whitehead's theorem (Theorem 3.3.17 in {{zb:0684.54001}}) the map $p\times\text{id}_K:\mathbb R\times I\to K\times I$ is also a quotient map.

And then later on, we only need to say: this induces a continuous function $F \colon K \times I \to K$ such that ... (i.e., the continuity follows from the the first paragraph).

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

[prabau]

For this proof I only have a quotient map p × id_I : ℝ × I → K × I, and how to induce a continuous map from K × I to K? I think the first paragraph of my proof that constructing ℝ × I → K is necessary, then using the property of quotient map.

I agree completely. What I was trying to say is that to show continuity of $F$, it's not enough to use the fact that $p$ is a quotient map. One also needs to use that p × id_I is a quotient map. That map is not mentioned in the proof.
Or maybe you mean that for algebraic topologists it's an obvious thing that does not need to be mentioned?

[prabau]

And yes, the first paragraph of your proof is absolutely necessary.

[prabau]

One of the commits for this PR shows as out of order with a commit date of Jul 30, 2038. Very strange.

[prabau]

will continue tomorrow

[prabau]

I think the description of the pseudocircle could be improved. The current one has no motivation.
I would describe it differently directly in terms of the Alexandrov topology on a certain poset.
Would you be ok if I directly add a commit for that? (We can discuss and modify afterwards as needed.)

[prabau]

Regarding the two bad commits that are messing things up (every time I write a comment, the comment disappears and then pops up at the bottom again after page refresh), would there be a way to fix this?
Maybe do a "squash" into a single commit and force push to github again?
Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

[yhx-12243]

Regarding the two bad commits that are messing things up (every time I write a comment, they pop up at the bottom again), would there be a way to fix this? Maybe do a "squash" into a single commit and force push to github again? Not sure how easy it would be, but you know more about git than me. Or is that a bad idea?

It seems a squash of whole push -f will solve it.

[yhx-12243]

Please do it, and I'll add my change afterwards.

Already done.

[prabau]

Please take a look. I also did some small rephrasing for a few other things.

The terminology "complete bipartite poset" is kind of intuitive. It is used in a few places, for example
https://dml.cz/bitstream/handle/10338.dmlcz/136284/MathSlov_32-1982-1_6.pdf

[prabau]

I need to ask you about simply connected.
What's the relationship with covering spaces?

(will continue tomorrow)

[yhx-12243]

I need to ask you about simply connected. What's the relationship with covering spaces?

(will continue tomorrow)

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

Or in details, we construct a loop $0 \to 1 \to 2 \to 3 \to 0$ and lift it to S51 (Khalimsky line).

[GeoffreySangston]

I think the pseudo-circle is the "non-Hausdorff suspension" over $S^0$ (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

[prabau]

For P200 (simply connected = false),

A simply connected space cannot have path-connected covering spaces other than itself, this is an easy theorem of algebraic topology.

I agree this is a basic result, follows from the homotopy lifting property. We just need to expand on what's there right now, and we can quote a result from Munkres for example. How about something like the following:

{S51} $K$ is a covering space of $X$ with covering map $p:K\to X:x\mapsto x \bmod 4$. The covering map has non-singleton fibers and {S51|P37}. Therefore $X$ is not {P200} (see for example Theorem 54.4 in {{ref to Munkres here}}).

[prabau]

I think the pseudo-circle is the "non-Hausdorff suspension" over S 0 (the 2 point discrete space); see Definition 3.4.2 of Finite Spaces and Larger Contexts; McCord's paper also has this and is much older. That seems to be a nice conceptual way of thinking about this.

Yeah, that's worth adding. I'll add a paragraph pointing to May-Pishevar. Not McCord as it is not accessible to me, but it's in the bibliography of May anyway.

[prabau]

I have added one more description of the pseudocircle and references.

@yhx-12243@GeoffreySangston Anything else you want to change or mention?

---

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

But it seems in the mathematical literature (from zbmath and Google scholar searches) pseudocircle is used with a different meaning: "A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum."
Not a problem here I think. If we ever need to disambiguate, it would be easy to do.

[GeoffreySangston]

Side comment: I don't have access to the McCord paper, but I assume it uses the term "pseudocircle". This was picked up by the wikipedia article, and from there it was repeated and used in multiple posts on mathse or mo.

McCord's paper doesn't use the term "pseudocircle". This space does appear implicitly in the paper, since he defines the non-Hausdorff suspension as in the other sources, and then applies it n-times to the 0-sphere, producing a space with 2n + 2 points. He highlights the case n = 2 at the very end of the paper, but also does not provide any special name for it.

[prabau]

Interesting. So this really was a made up term in Wikipedia and it took a life of its own.

[hew-wolff]

Thank you for adding such a nice entry on this!

By the way, if you want to see the McCord paper (I"Singular Homology Groups and Homotopy Groups of Finite Topological Spaces", Michael C. McCord), someone put it up at https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf.

"Finite Spares and Larger Contexts" by May and Pishevar is a helpful reference, but the copy I found is a bit raw: there are gaps and marginal notes indicating that it's not formally published. Another good source is "Algebraic Topology of Finite Topological Spaces and Applications" by Jonathan A. Barmak (Springer, 2011), which is at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/barmak2.pdf. It does not use the term "pseudocircle", but the construction is clearly treated in section 3.1, "A Finite Space Approximation".

[prabau]

Thanks. Barmak's book seems very interesting.