# Tags for separation axioms and countability axioms for topological spaces

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 and 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 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 $\to$ , 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 or 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 and tags. (Both of them are still small enough, if we want to change the tag names.)

I will start retagging the questions tagged 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.

• I think that separable spaces should be promptly synonymized with countability-axioms. There's no reason to separate them. Similar reasoning indicate that synonyms for separation axioms related tags. The tags do not form a Hausdorff space; and there shouldn't be any separation between some of them. – Asaf Karagila Mod Dec 8 '16 at 18:00

I don’t think that there should be a 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 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 tag is fine, and if not, a Hausdorff tag is inappropriate.

• So to summarize, current state is that we have these tags: (separation-axioms), (countability-axioms), (separable-spaces). Your proposal would mean removing (countability-axioms) and possibly also creating new tags for first countable and second countable spaces. (In the process of tag-removal, those questions could be retagged by a more appropriate tag.) Is that about right? – Martin Sleziak Dec 9 '16 at 13:13
• @Martin: Yes, exactly. – Brian M. Scott Dec 9 '16 at 13:15
• @Martin: I'd think that synonymizing is more appropriate. People who study their first course in topology don't usually get exposed to these cardinal functions, but just to their "countable case". So having tag synonyms is generally a good idea here. – Asaf Karagila Mod Dec 10 '16 at 9:01
• I would caution, however, against a [cardinal-functions] tag. As much as I would have been happy to see one, it will be wildly misused by people asking about cardinal arithmetic and the likes of it. – Asaf Karagila Mod Dec 10 '16 at 9:04
• @AsafKaragila I will think about this for a bit more before I actually upvote one of the answer, but this one seems rather reasonable to me. I would agree that first countability, second countability and separability are probably important enough to deserve a separate tags. (Although some quid's remarks both here and in chat make me think whether we could think of a better name than (separable-spaces). I mean the part where they recommend not to name them like the objects, although it was not in connection with separability.) – Martin Sleziak Dec 10 '16 at 9:12
• This answer actually does not suggest creating (cardinal-functions) tag. If such tag is ever considered, I would expect name like (cardinal-functions-topology) somehow shortened to fit the 25 characters limit. But this is for another discussion, if the need for such tag arises in the future. – Martin Sleziak Dec 10 '16 at 9:12
• @Martin: In the abstract a cardinal functions tag is a good one, but there’s no evidence that we actually need it. There have been a few questions that would definitely have fallen under it, but only a few. If it ever is needed, it should probably apply both to topology and to Boolean algebra: Don Monk has a couple of books on cardinal functions on Boolean algebras, and a lot of them correspond closely to the topological functions. – Brian M. Scott Dec 10 '16 at 19:53
• @Martin, Brian: I didn't say that a tag for cardinal functions is in order. I just pointed out that in case that such thought goes through anyone cranium, then it is worth noting that in all likelihood, a tag named [cardinal-functions] is going to be heavily misused. Not that it's necessarily a bad idea, but currently it's still in a very theoretical place, and it doesn't have enough traction as far as empirical evidence goes. – Asaf Karagila Mod Dec 11 '16 at 22:40
• @AsafKaragila To get back from this digression from the original topic to cardinal functions, I will add that also the tag (countability-axioms) is misused for questions about countable sets, one example. So removing the tag might mean one problem less - although some users with probably misuse the tags (first-countable) and (second-countable) in a similar way. (Or whatever will the tag for first countability or second countability be called, if they are created.) – Martin Sleziak Dec 12 '16 at 0:27
• @Martin: There is no escape from the misuse, but countability axioms deserve tag(s). – Asaf Karagila Mod Dec 12 '16 at 2:32
• So if the tags for first- and second-countable spaces are created, what would be a reasonable tag name? I guess both (first-countable) or (first-countability) sound reasonable. And both (first-countable-spaces) and (second-countable-spaces) still fit 25 characters limit for tag name, but they seem a bit too long. – Martin Sleziak Dec 12 '16 at 9:14
• @Martin: Admittedly, I'm not entirely sold that those deserve separate tags; or on Brian's view that countability-axioms is not a very good tag name. That being said, I'm not sure that it is a good tag name, and that we can't do better. – Asaf Karagila Mod Dec 12 '16 at 13:16

On the last question: no there should be no synonym from to .

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 . 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 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 . 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 or rather than . As explained above the point is to avoid them being used for questions where the objects merely happen to appear.

• It will always be a judgement call whether notion that appears in that question is relevant enough to be used a tag. (We all have seen repeated comments by Asaf saying: If the question contains word set, that does not mean it is (set-theory).) However, if the tag-info explicitly says that that (countability-axioms) is for question about separable spaces, I do not see that much of a problem. If some user is unsure why the tag is there, they can simply hover over it. – Martin Sleziak Dec 9 '16 at 5:06
• To be completely honest, originally separable spaces were mentioned only in the tag-wiki, now I have added it into the tag-excerpt too, so that this information is more visible (shown in the tooltip). – Martin Sleziak Dec 9 '16 at 5:08
• Do you think that somebody looking for questions about countability-axioms will be glad to find the question I gave as example? I doubt it. Therefore it should not be tagged like this. I see two options: either 'kill' the tag separable spaces, or keep it with a remark that if appropriate the additional tag countability-axioms should be added. A synonym though is not desirable. (Generally, I feel that many but the most direct synonyms cause more harm than good.) – quid Mod Dec 9 '16 at 10:22
• The way I understand current tag-info for (countability-axioms), the tag (separable-spaces) is subtag of this tag. (AFAICT it was used exactly in this way - as the tag where either tag (separable-spaces) of (first-countable) or... would have been used, if they existed.) So every question tagged (separable-spaces) should be also tagged also (countability-axioms) - at least according to current tag description. So if you are pointing out an example where you think (countability-axioms) would be inappropriate, then it is also incorrectly tagged (separable-spaces). – Martin Sleziak Dec 9 '16 at 10:36
• "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." – quid Mod Dec 9 '16 at 10:50
• On the other hand, if there is a synonym say (hausdorff-property) $\to$ (separation-axioms), in some cases it will help tagging. Probably a new user will be unaware that if their questions is about Hausdorff spaces, it should be tagged (separation-axioms). But when they start typing Hausdorff in the tag field, they might see the synonym. But since this is getting too long, let's either agree that we disagree on this, or let us continue the discussion in chat. – Martin Sleziak Dec 9 '16 at 11:00
• It is not clear to me whether you change from "space" to "property" on purpose in your comment. I'd have no problem with h-property being a syn, indeed I more-or-less proposed this. What I am against is making common names of objects a synonym of something, because this leads to mis-tagging and "pollution" of the target, with questions where the objects just happens to appear. Whether or not the tag should (in some abstract sense) be use is only that relevant; as a matter of fact the tag will be used, and it is also hard to avoid as for the asker it often will not be clear it is tangential. – quid Mod Dec 9 '16 at 20:21
• I have said (hausdorff-property) since it was one of the possibilities you have mentioned. I should probably read your post more carefully, since my impression was that you are against synonyms of the form (hausdorff-something) $\to$ (separation-axioms) in general. (Here something stands for the name of the tag which would be chosen.) I am glad that this was clarified in your comment. (I have added a footnote in my question as a reaction to the above comment.) – Martin Sleziak Dec 10 '16 at 5:34