Adds two corollaries of Mansfield paper

[Moniker1998]

Closes #1568

[Moniker1998]

I've changed my answer a bit, the previous version contained an error.

In particular now the theorem for ordinals is a corollary of the proof of the theorem in Mansfield's paper, rather than corollary of a theorem itself. In the sense that the more general statement can be proven in the same way.

[prabau]

T821, 822: Would be convenient to add Mansfield to the refs, with text like this for example:
See {{mathse:...}}, based on results from {{zb:0078.14803}}.

[Moniker1998]

@prabau

I've changed it, but not exactly how you wanted since it just isn't true that both of these theorems are entirely based off of results in Mansfield. One is a generalization of a proof, and the other also uses a result of Stone (I believe) in Lutzer.

[prabau]

I completely agree that your mathse proof adds more that what is in Mansfield. For example, for T821, you show in full generality which ordinals are paracompact based on their cofinality. That is very nice, whereas Mansfield only deals with successor ordinals.
That's why I had suggested "based on Mansfield", which means it's making use of results of Mansfield, not that it is an exact repetition of results of Mansfield.

Anyway, the current wording is not so good.

T822: "This was established for LOTS in Corollary 3.6 of {{zb:0078.14803}}."
That is not good. Corollary 3.6 is phrased in term of uniformities, which pi-base's P207 never mentioned. That is confusing for users of pi-base. No need to mention that corollary. What you have in the mathse post, based directly on Thm 3.4 and not Cor 3.6, is perfect, direct and clear.

T821: We should revert to have "paracompact" and not "fully normal" in the hypotheses. It's equivalent for T4 spaces, so for ordinals in particular. And "paracompact" is more well-known to people than "fully normal". It's also nominally a stronger result to use a weaker hypothesis (although, as mentioned, it's equivalent in this case).

Also, the wording is awkward: "See Math StackExchange 5117210 for generalization of Theorem 3.3 in zbMATH 0078.14803, which this result is a Corollary of."
Your mathse explanation is good, clear, and not a corollary of Thm 3.3. Instead, it refers to the method of proof of Thm 3.3, as well makes use of Lemmas 3.1 and 3.2. But no need to mention all that in the pi-base text.
Furthermore, the full mathse post 5117210 does not mention anything related to this result. This result only appears as an addendum to the mathse answer. So we should refer to the "Answer to ..." instead.

Suggestion: change the ref to mathse:5117290 (Answer to "..."). And change the text to this for example:
See the second part of {{mathse:5117290}}, making use of results of {{Mansfield}}.

[Moniker1998]

I completely agree that your mathse proof adds more that what is in Mansfield. For example, for T821, you show in full generality which ordinals are paracompact based on their cofinality.

Yes, even more than that, since I analyze their $\aleph_\alpha$-full normality.

That is very nice, whereas Mansfield only deals with successor ordinals.

Successor cardinals, not ordinals.

T821: We should revert to have "paracompact" and not "fully normal" in the hypotheses. It's equivalent for T4 spaces, so for ordinals in particular. And "paracompact" is more well-known to people than "fully normal". It's also nominally a stronger result to use a weaker hypothesis (although, as mentioned, it's equivalent in this case).

What is proven are results related to full normality. It would be unnatural to mention paracompactness. As you say, results on pi-base show the two are equivalent for any GO-space.

Suggestion: change the ref to mathse:5117290 (Answer to "..."). And change the text to this for example:
See the second part of {{mathse:5117290}}, making use of results of {{Mansfield}}.

Yes.

[prabau]

T821: We should revert to have "paracompact" and not "fully normal" in the hypotheses. It's equivalent for T4 spaces, so for ordinals in particular. And "paracompact" is more well-known to people than "fully normal". It's also nominally a stronger result to use a weaker hypothesis (although, as mentioned, it's equivalent in this case).

What is proven are results related to full normality. It would be unnatural to mention paracompactness. As you say, results on pi-base show the two are equivalent for any GO-space.

It is perfectly natural to present the result in terms of paracompactness. How a result is proved does not matter here. What matters mostly to users of pi-base is the result itself. Repeating what I said earlier, "paracompact" is more well-known to people than "fully normal". It's also nominally a stronger result to use a weaker hypothesis (although, as mentioned, it's equivalent in this case). So phrasing the result as [ordinal + paracompact => sigma-compact] is a very good way to present things.

So, where your post says "it follows that for ordinal spaces σ-compact and fully normal are equivalent", you can add a sentence mentioning the same with paracompact. And everyone will be happy.

@yhx-12243 @felixpernegger fyi

[Moniker1998]

It is perfectly natural to present the result in terms of paracompactness.

It isn't.

How a result is proved does not matter here.

I think it does matter what is being proven.

Repeating what I said earlier, "paracompact" is more well-known to people than "fully normal".

I don't see why that matters.

So, where your post says "it follows that for ordinal spaces σ-compact and fully normal are equivalent", you can add a sentence mentioning the same with paracompact. And everyone will be happy.

I think one usually says "And everyone will be happy." when they present some sort of compromise. I don't see any presented.

[prabau]

I think one usually says "And everyone will be happy." when they present some sort of compromise. I don't see any presented.

The compromise is that if you add to the mathse post the explanation that for ordinal spaces paracompact is equivalent to sigma-compact, you will have proven that fact. And therefore we can say that whatever we provide as proof of a result has indeed a proof of that result, which is what you wanted.

[Moniker1998]

The compromise is that if you add to the mathse post the explanation that for ordinal spaces paracompact is equivalent to sigma-compact, you will have proven that fact.

And how is that a compromise?

And therefore we can say that whatever we provide as proof of a result has indeed a proof of that result, which is what you wanted.

I don't know what you mean here. I don't think I currently want anything like you are describing. Not sure what this is referring to.

[prabau]

I don't know what you mean here. I don't think I currently want anything like you are describing. Not sure what this is referring to.

See #1570 (comment): "What is proven are results related to full normality. It would be unnatural to mention paracompactness."

Anyway, it does not matter too much. So approved.