Tags for separation axioms and countability axioms for topological spaces

Question

In topology, several separation axioms (such as Hausdorff, completely regular, normal, ...) and countability axioms (such as first countability or separability) are often used ant there are also some questions about them on this site. Hence it is not entirely surprising that we also have tags separation-axioms and countability-axioms for such questions. And also tag-info for both separation axioms and countability axioms have been created. Both these tags have been around for some time, the tag for separation axioms was created in September 2013, the tag for countability axioms was created in February 2016.

However, I think that creating separate tags for each one of them might lead to having too many tags. (Just consider the fact, that many questions might be about implication between two separation properties, so such question would quite naturally be tagged with both.) This is perhaps less clear for countability axioms, but there are certainly many separation axioms.

This questions was prompted by the fact that the tag separable-spaces has been created not so long ago. (Apart from this one, I do not know about a tag for single one of these properties.) However, since these properties are similar, perhaps it makes more sense to discuss them all at once rather than each of them separately.

• Should we have separate tags for some of separation axioms? Should we have separate tags for some of countability axioms? (If yes, for which ones and what would be reasonable names for those tags be?)
• Are there some properties which should be added to the tag-info, so that it is clearer which topological properties are covered by these two tags.
• Should we create synonyms such as¹ hausdorff-spaces $\to$ separation-axioms, so that this helps users who are looking for tag when asking about Hausdorff spaces (and similarly for other properties)? Or does it suffice to create synonyms only in cases when some user creates such tag?

¹Perhaps a better name would be hausdorff-property or hausdorff-axiom for the reasons explained in quid's post. However, this was merely given here as an example - as we are already discussing these tags, we might discuss such possible "redirects" at the same time. (Both whether they might be useful and if yes, what could a good choice of name be.)

EDIT: Since there was no activity on this post for some time, I have accepted the most upvoted answer. As a result of this discussion, I have created first-countable and second-countable tags. (Both of them are still small enough, if we want to change the tag names.)

I will start retagging the questions tagged countability-axioms in a reasonable pace until we eventually remove the tag. List of the questions which had this tag before the retagging started can be found here.

Answer 1

I don’t think that there should be a countability-axioms tag in the first place: it’s not a natural category. Second countability, first countability, and separability are just names for the countable cases of the cardinal functions weight, character, and density, respectively. If we do have it, it should include spaces of countable spread, tightness, extent, height, depth, cellularity, and so on through the list of standard cardinal functions. If we’re going to have any tags in this area, they should be restricted to specific, frequently occurring countability properties, and first countability, second countability, and separability are the obvious choices. But they really have nothing in common beyond the fact that they specify the smallest value of a useful cardinal function, so there should not be a cover tag for them.

I take exactly the opposite view of separation axioms: the single tag separation-axioms is the way to go. I’m perfectly happy to let it cover questions about the $T_1$ property, the Hausdorff property, normality, the Tikhonov property, etc. I see no benefit to having a separate tag for Hausdorff spaces, for instance: if the Hausdorff property is actually usefully relevant, a separation-axioms tag is fine, and if not, a Hausdorff tag is inappropriate.

Answer 2

On the last question: no there should be no synonym from hausdorff-spaces to separation-axioms.

The issue is that somebody asking about a problem that happens to be placed in the context of a Hausdorff space might well use the former tag, while the question is not at all about separation axioms.

Now, one might argue the question then should not be tagged Hausdorff space to begin with, but that might be arguable even in theory and anyway in practice users just tag like that, and we should try to avoid synonyms that are likely to turn so-so tagging into tagging that makes no sense.

The same applies to separable-spaces. An existing question with that tag is a good example for the risk: Let $\mathcal{H}$ denote a separable Hilbert space. Construct a nonzero operator $T \in B(\mathcal{H})$ such that $T^2 = 0$.

To tag this separable-spaces might not be needed or optimal tagging, but it is transparent why it was done and it makes some sense. Now, imagine it was tagged countability-axioms. It would make little sense to me.

I do not know if specific tags are needed. But if they are to be created, I'd recommend, at least in some cases, not not to name them like the objects. That is, one might use hausdorff-property or hausdorff-axiom rather than hausdorff-space. As explained above the point is to avoid them being used for questions where the objects merely happen to appear.