An update to S57 (rational sequence topology)

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?

[pzjp]

Some missing traits, zbMath references, cleanup.

I suspect that the traits missing at the moment may depend on the particular construction.

[pzjp]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare $\aleph_1$-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable $\Rightarrow \lnot T_4$

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

[yhx-12243]

Actually Jones' lemma makes it worth to add a property has (at least) continuum extent. Actually most spaces with uncountable extent have it equal continuum. In the rare ℵ₁-extent-cases we will end up with ZFC-independent trait. Then Jones' lemma could be added:

has continuum extent + separable ⇒ ¬T₄

Probably this could be strenghtened to not Normal but never seen the non-Hausdorff version.

Great. This is in fact an important corollary of Tietze extension theorem (Corollary 2.1.10 in Engelking).

Also, “has continuum extent” appears at an other valuable result:

Has continuum extent + separable ⇒ ¬countably paracompact.

See the claim in https://www.sciencedirect.com/science/article/pii/S0166864106001520.

[prabau]

By "has continuum extent", did you mean "has extent $\ge\mathfrak c$ " = "has extent at least continuum" ?

[pzjp]

Yes. Skipped the "at least" part only to shorten the text. But I guess the eventual official name should be precise.

[prabau]

P7: Jones's lemma holds for normal spaces, not just for T4 spaces. So it seems better to show directly that the space is not normal (P13).

We don't need to reprove Jones's lemma. Reference: Willard Lemma 15.2. and then just apply it: $X$ is separable and $X\setminus\mathbb Q$ is closed and discrete and of cardinality $\mathfrak c$. Hence not normal.

[prabau]

P132 ($G_\delta$ space) is a consequence of scattered (P51) plus developable (P110).

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first.
Marking this one as draft and will review #1393 tomorrow.

[pzjp]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

[prabau]

Just realized: We have to review #1393 first, so that the suggested theorem T756 becomes available first. Marking this one as draft and will review #1393 tomorrow.

The space won't be completed before T756 but this PR does not explicitely depend on the latter. We can change the title to "almost completing" in order to be precise.

The reason I marked it as draft is not for that. It's because we used to have P82 (locally metrizable) for this space, but we cannot conclude this anymore without T756.

The better way would have been to have T756 in its own PR, so it could have been reviewed and merged quickly. And then committing this particular PR would not have lost any trait.
Or alternatively, not remove P82 in this PR, and then remove it later.
(Good to know in general, to be aware of dependencies, but here it probably does not matter much.)

Should we finalize this and merge anyway?