@ArthurFischer created the-baire-space when editing this post.

The tag might be useful. However, I wonder whether descriptive-set-theory had to be removed.

In any case, many questions about Baire space that we have on this site at the moment are probably tagged (descriptive-set-theory).

Maybe I should have phrase it like this: Should (the-baire-space) be a subtag of (descriptive-set-theory)?

I have first encountered this space in connection with DST. And despite the fact that it is metric (metrizable) space and we can speak about it just considering the metric structure/topological structure; in my mind it is still closely connected to DST.

off-topic question: Did you receive ping from this room or did you simply notice that something was posted here, Arthur? (I'd guess that you were not pinged, since your name was not offered me in the auto-complete using tab.)

I wonder how long will it take before somebody (Asaf?) will make a post that this tag is too specific. (Similar to the above posts about Cantor set.)

@MartinSleziak I noticed a couple edits from you, and was certain I'd find some new content here. You're pretty predictable that way. ;-)

@MartinSleziak Here are my thoughts. The d-s-t tag should be used for questions really about descriptive set theory: definability (Borel, projective, etc.) of subsets, uniformazation, separation theorems, etc. While the Baire space is central to d-s-t, not every question about it is really a question of d-s-t. That it is not sigma-compact is a purely topological property of the space, and doesn't really figure directly into d-s-t, as I (probably mis-)understand it.

I thought so. So, I assume, ping only works for names which can be auto-completed using tab key.

You are certainly correct that it is topological question.

Based on what you wrote, probably none of the 4 questions currently tagged the-baire-space should have descriptive-set-theory tag.

I'll try to explain my reasoning on some examples, if I can think of some.

Stone-Cech compactification is studied in topology. If we had (stone-cech-compactification) tag it would not be unreasonable to expect that every question having this tag would also have (general-topology) tag.

But I can imagine question about $\beta\omega$, which is mostly about semigroup structure or about some properties of ultrafilters, where topological properties would not play a big role.

If the Baire space is almost exclusively studied in descriptive set theory, then the situation is similar.

But this is (probably) not the case.

To be more precise, the correlation between baire-space and descriptive-set-theory is much weaker than the correlation between stone-cech-compactification and general-topology.

So basically I have no objections against your assessment. But I thought that it is good that it was discussed at least for a bit.

As I have mentioned, many questions about the Baire space are currently under (descriptive-set-theory) tag: math.stackexchange.com/…

So if we are going to retag at least some of them, we can also have a look whether d-s-t tag should be removed or not.

Speaking of topics related to descriptive set theory, should we have polish-space tag?

I have removed descriptive-set-theory from some question tagged the-baire-space (and also added metric-spaces in some cases.)

@MartinSleziak I'm pretty sure you will find the Baire space (not so-named) as an example in general topology texts. But I agree that it is more important to d-s-t (where it plays a pretty central role) than to general topology.

@MartinSleziak I think Asaf had a role to play in a lot of these...

Isn't there even a name for the particular metric which is most frequently used to for $\mathscr N$?

@MartinSleziak I'm not so sure. I really don't know how many questions are out there about Polish spaces (as opposed to specific Polish spaces).

It's Baire metric? (IIRC Frechet metric is something different.)

I don't recall off the top of my head.

@ArthurFischer You're probably right. Probably polish-space is not needed.

We will see what happens with the-baire-space. In any case, if I see new questions I will use this tag. And I will try to retag a few older ones from time to time.

Just for the record (and so you have something to link to when the time comes), I decided to create this tag because that space is used under so many different names/descriptions that it is virtually impossible to find duplicates. Add to this that searching for MathJax is terrible on this site, and there's probably an entire corpus of questions/answers that no-one can find.

We have some questions about Polish spaces, but it is probably better to use descriptive-set-theory than to create a separate tag.

Thanks for your explanation of the reasons for creating the new tag. (As you say, it might be useful ot have something to link to in case this is discussed. And perhaps also for discussions about some similar tags.)

Although there is also an older discussion in connection with [tag;sorgenfrey-line], where you have mentioned something similar.

Only now I've noticed that the explanation when to use descriptive-set-theory together with the-baire-space is given in the tag-wiki.

I will also add a link to the tag-excerpt.

Would convergence-tests tag for questions about convergence tests be useful? Perhaps as a subtag of sequences-and-series.