$\newcommand{\mc}{\mathcal}$ $\newcommand{\set}[1]{\{#1\}}$ Notation and Definitions Let $G=(V, E)$ be a graph. For $X, Y\subseteq V$, we write $e_G(X, Y)$ to denote the number of edges in $G$ with one end-point in $X$ and the other end-point in $Y$; the edges with both end-points in $X\cap Y$ ...

For a topological space $X$ that is Hausdorff, and $\{B_k\}_{k\in I}$ is an arbitrary collection of compact sets, I need to show that $\bigcap\limits_{k\in I} B_k$ is also compact. Here's the rub. I do not have closed sets to work with. The proof that compact sets in a Hausdorff space are clos...

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 sep...

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, ...

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 axiom...

I've noticed we have both legendre-symbol and quadratic-residues. Would it make sense to make the former an alias of the latter?