Theorem Suggestion: Two theorems about GO-spaces and ordinal spaces

[Moniker1998]

Theorem Suggestion

1. GO-space implies strongly collectionwise normal
2. Paracompact + ordinal space implies $\sigma$-compact

Proof/References

I've wrote it up pretty well here.

[felixpernegger]

Thanks hahaha, this is really great

[felixpernegger]

The first result is definitely suitable.

The second result does not actually have any new implications (didnt check if it gives old manual traits automatically though), but it still seems good.

Can you make the PR? Maybe its good to put the papers you mentioned on MSE directly in the references tab

[Moniker1998]

Maybe its good to put the papers you mentioned on MSE directly in the references tab

No because those aren't direct results. It actually takes some thought to connect the two and two together and get the results we want, however trivial it wouldn't be.

[felixpernegger]

Have you checked the proofs yet by the way?

[Moniker1998]

Depends on which one do you mean.

That LOTS are $\aleph_0$-fully normal and the result for closed subspaces I haven't read before.
The result about GO-spaces being closed subspace of LOTS I've read. (today in fact)
Cohen's theorem I've also read I believe, because I needed it for something about strongly collectionwise normal spaces.
I haven't read why $\omega_{\alpha+1}$ are not fully normal either.

It would be just these three results that I haven't really read into.

Why, do you want me to review this? It would be faster I guess since I'm already into this a little.

[felixpernegger]

No worry, I just wondered. Just because I think when we use outside papers it definitely suffices for 1 person (i.e. the author) to read it

I can read the other stuff as well of course if you want me to, since I originally wondered about it

[Moniker1998]

I don't think these results would be particularly hard and double checking would definitely be better
Assuming there isn't some kind of deeper background needed to understand them

[felixpernegger]

Alright

[prabau]

A general comment. When a proof of something involves some notion that are not (yet?) in pi-base, it's worth asking ourselves if it would be beneficial to add this as a new property, instead of just referencing a bunch of results making use of that notion.

In this case, is the notion of $\aleph_0$-fully normal worth having in pi-base? Is that something that has appeared in the literature in more places than just that paper, maybe with different names? Does it have interesting connections to other properties? Would it be worth having in pi-base, as it would allow some fined grained distinctions between spaces? Or is it just too specialized and not of much interest except as an intermediate notion to prove some other result?

(I am not suggesting we should add this property, as I have not even looked at it. Just asking you if you think it would be worth adding.)

[Moniker1998]

I don't think it would be worth adding.
There is not just one property here, and I don't see how they would connect to other properties in any way.

Namely, $\aleph_0$-fully normal, finitely-fully normal, $2$-fully normal, almost $\aleph_0$-fully normal, almost finitely fully-normal, almost $n$-fully normal for $n\in\omega$, is all properties we could be looking at.

And while they are interesting in that they are various weakenings of the notion of fully normal that allow us to grade spaces, there does not seem to be any obvious connections or reasons to prefer one over other.