 5:13 AM
The questions below are all about the standard definition if integers from natural numbers, i.e. we have equivalence relation $(x,y)\sim(x',y')$ defined by $x+y'=x'+y$. Then addition and multiplication is defined in the natural way.
All questions below are about the question whether multiplication is well-defined.
They are tagged by all sorts of tags.
What would the correct tags be? I think this is one of rare occasions when the fit. This is about construction of $(\mathbb Z,+,\cdot)$. And I would also be tempted to use - since the question is whether some function is well-defined, i.e., how it behaves on equivalence classes.
0  I'm having a bit of trouble on this proof. It's part of the construction of the integers. $R$ is the relation, $\mathbb{N}$ the natural numbers, $((x,y),(n,m)) \in (\mathbb{N}\times\mathbb{N})\times(\mathbb{N}\times\mathbb{N}) | x + m = y + n$. $(x',y')\in[(x,y)] = x'+ y = y' + x$ $(n'm')\in[(n... 3  Let$E$be an equivalence relation on the set of all ordered pairs of non-negative integers ($N\times N$). It is defined as $$(a,b)E(x,y) \Longleftrightarrow a+y = b+x$$ Multiplication ($*$) is defined as $$(a,b)*(x,y) = (ax+by, ay+bx)$$ Without using substraction or division, how can I show th... 1  We will define the relation ~ on$\mathbb N \times \mathbb N$by$(a,b)\sim (c,d)$iff$a + d = b + c$. Prove that the operation given by:$[(a,b)][(c,d)] \stackrel{\text{def}}= [(ac + bd, ad + bc)]$is well-defined. My attempt at this proof: Let$(a,b)$and$(a',b')$be elements of$[(a,b)]$,... 1  If$(a,b) \sim (A,B)$and$(c,d) \sim (C,D)$, where all pairs are whole numbers, Prove that:$(a,b)*(c,d)\sim(A,B)*(C,D)$Relation defined on the following:$(a,b)\sim (A,B) \iff a+B=A+b(c,d)\sim(C,D) \iff c+D=C+d$For this I am assume we have to work a little backwards to show the proof... Apart from retagging, they might need also other improvements. (Are titles descriptive enough? Should some of them be closed as duplicates? 6 hours later… 10:55 AM 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 6 hours ago 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 6 hours ago 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 33 mins ago 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 19 mins ago 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 8 secs ago
To be honest, I still do not see what is the big difference between a question which is incorrectly tagged and a question which is incorrectly tagged .
If the community consensus (or guidelines for usage of that tag) say that simply stand for (and a few other tags), then it is exactly the same thing.
For example, if a month ago, when did not exist, if a user would like to search for questions about them using tags, they would use .
And sorry if I am being too obnoxious about this. I can see your point, but I have a different opinion. And if some kind of consensus is reached there, I will certainly do my best to follow it. 11:21 AM
I will just add that I am not pushing for synonym for Hausdorff spaces. I am just saying that I can see situations when it will improve tagging. And it is likely that in some situations it would not help.
More important question what to do with (separable-spaces) tag - synonymize, keep separate or remove - since the tag already exists. But I thought it is better to discuss it more generally in one question than discussing tag later in a separate post if somebody creates it.
In the case of this tag, it seems that Asaf is for synonym:
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 17 hours ago

1 hour later… 12:26 PM
I don’t think that there should be a countability-axioms tag in the first place: it’s not a natural category. The relevant concepts are weight, character, and density: a second countable space is one with countable weight, a first countable space is one with countable character, and a separable space is one with countable density. 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 ... — Brian M. Scott 20 mins ago
... 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. — Brian M. Scott 19 mins ago
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 normality, complete regularity, etc. I see no benefit to a tag for Hausdorff spaces, say: if it’s actually usefully relevant, so is separation-axioms. — Brian M. Scott 12 mins ago
@BrianM.Scott I really appreciate that you joined the discussion. Since you are very active in questions from general topology, your opinion certainly has some weight here. It would be nice if you could post your comments as an answer, so that other users can vote of them. To save you from retyping what you already posted as comments, you can find them here. — Martin Sleziak 1 min ago
4  In topology, several separation axioms (such as Hausdorff, 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 c...

Since we are having a discussion about and tag, we might as well check when they were created.
At least I can test arjafi's queries from here: Can you tell who created a tag
For separation-axioms the query returns this question, the tag was added there by MJD in September 2013.
Not so long after that I have created tag-info: chat.stackexchange.com/transcript/3740/2013/9/9
For countability-axioms the query returns this question, if that is the first occurrence of the tag, it was created by learnmore. If was in Februrary 2016.
The first mention of that tag in this room was around the same time: chat.stackexchange.com/transcript/3740/2016/2/12
In both cases, I have pinged probable tag creatyor.
@MJD Since you are probably the user who created (separation-axioms) tag, I thought it might be useful to let you know that there is an ongoing discussion on meta related to this tag. — Martin Sleziak 1 min ago
@learnmore Since you are probably the user who created (countability-axioms) tag, I thought it might be useful to let you know that there is an ongoing discussion on meta related to this tag. — Martin Sleziak 58 secs ago
@Martin: Done, so I’ve gone ahead and deleted the comments. — Brian M. Scott 10 mins ago
0  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, ...

1 hour later… 1:51 PM
Since the possibility of the removal of (separation-axioms) has also been suggested, I will give here links to revision history for tag-excerpt and tag-wiki. (So that they are still easily accessible if the tag is deleted.) 2:06 PM
@MartinSleziak;Thank you very much;I did have a look at it — learnmore 8 mins ago
There are some questions which use (countability-axioms) tag simply as: "Well, do we have some tag about countable sets? This seems like that." They should be retagged no matter what the outcome of the discussion on meta is. The questions tagged countability-axiom but not general-topology are natural suspects.
If after that we have, say, 30 questions in tag, the manual removal is certainly feasible. (We did much large retaggings than this in the past.)

3 hours later… 4:47 PM
\begin{equation}
E=m
\end{equation}
\begin{equation}
testing latex..\pi \theta
\end{equation}
$testing latex..\pi \theta$

4 hours later… 9:10 PM
@MartinSleziak the example I gave is not really incorrectly tagged separable spaces. This is the difference.
@MartinSleziak that's be a poor naming. I see that the wiki somewhat suggests this, but I do not think it was really intended as such. Instead the wiki summary is more telling.
Further and that's the actual problem, once a tag separable-spaces exists there might be more 'incorrect' usages of that tag. Either one should make it so that they are kind of correct or disallow the tag.
Different usage also have slightly different connotations. Another example is the min-max thing. I think it is not a good idea to have this under optimization.
A student asked to find (local) maxima and minima of x^5 - x^3 is not really doing optimization.
re propoduct-spaces: my concern is somewhat different. A question "For fin dim vector spaces, show that dim V x W is dim V + dim W" is also a question about product spaces. I do not think the tag should be used. The wiki allows it though. Maybe the "for example" should be changed, and tag be actually restricted to top and measure theory (conceived broadly).