Is there any intuition behind separation axioms being in terms of closed sets

Question

It feels like there should exist some intuition behind this. Additionally, Muckres, our textbook, isn’t big on intuition or motivation which lends credence to the conjecture that there could exists so intuitive explanation.

Answer 1

I'd like to provide another motivation: To supply enough continuous real functions.

We say a family $\mathscr{F}$ of continuous functions on a space $X$ separates points from closed sets iff for every closed $F\subseteq X$ and $x\in X-F$, there exists $f\in\mathscr{F}$ such that $f(x)$ is not in the closure of $f[F]$. Now consider the following theorem:

If $\{f_i\colon X\to X_i\}_{i\in I}$ be a family of continuous functions that separates points from closed sets in $X$, then $X$ has the initial topology induced by $\{f_i\}_{i\in I}$. Moreover, if $X$ is $T_1$, then the evaluation map $\epsilon$ from $X$ to the product space $\prod X_i$ defined by $$ \epsilon(x)=(f_i(x))_{i\in I} $$ is an embedding.

This theorem is most interesting when every $f_i$ is a bounded continuous real function. In this case, we can study $X$ as a subspace of a product space of bounded closed intervals (called a cube). This technique can be used to prove important results like Urysohn's metrization theorem, the existence of Stone-Čech compactifications, etc.

So we'd be happy if a $T_1$ space $X$ admits a family of bounded continuous real functions that separates points from closed sets (boundedness is no real restriction here). That is, for every closed $F\subseteq X$ and $x\in X-F$, we want a continuous real function such that $f(x)$ has an open neighborhood $(f(x)-r,f(x)+r)$ disjoint from $f[F]$. In fact, it suffices to find continuous $f\colon X\to [0,1]$ such that $f(x)=0$ and $f[F]=\{1\}$ (that is, it suffices to show that $X$ is $T_{3\frac{1}{2}}$). Indeed, if $(g(x)-r,g(x)+r)$ does not intersect $g[F]$ for some continuous $g\colon X\to\mathbb{R}$, then we may assume $g(x)=0$ and $r=1$ without loss of generality, and define $f\colon X\to[0,1]$ by $$ f(x)=\min\{|g(x)|,1\}. $$ Then $f\colon X\to[0,1]$ is a continuous function that maps $x$ and $F$ to $0$ and $1$ respectively (for convenience, we say $f$ separates $x$ from $F$).

However, given arbitrary closed $F\subseteq X$ and $x\in X-F$, it can be quite difficult to find a continuous real function $f$ that separates $x$ from $F$. A necessary condition for such $f$ to exist is that $x$ and $F$ are separated by some disjoint open sets, because $\{f<1/2\}$ and $\{f>1/2\}$ are such open sets. Such a $T_1$ space is said to be $T_3$. Since topologies are defined in terms of open sets, it is often easier than $T_{3\frac{1}{2}}$ to check if a space is $T_3$.

Unfortunately, not all $T_3$ spaces are $T_{3\frac{1}{2}}$ (the counterexamples, however, are very difficult to construct. See this post on MO). That's when $T_4$ comes into the picture: By Urysohn's lemma (which is really hard to not appreciate), every $T_4$ space is $T_{3\frac{1}{2}}$, and $T_4$ is defined in terms of open sets.

I'd say $T_2$ (which ensures the uniqueness of limits) and $T_4$ are the most important separation axioms. But $T_4$ spaces behave badly in some other ways. Unlike $T_n$ for $n\leq 3\frac{1}{n}$, being $T_4$ is not preserved by taking (non-clsoed) subspaces and products, making it harder to produce new $T_4$ spaces from existing ones. For arbitrary subspaces, there's the $T_5$ axiom (enjoyed by all linearly ordered spaces). For products of even just two normal spaces, as far as I know the study can go quite deeply into set theoretic topology (see Dowker space).

Answer 2

To expand on Andreas's example in the comments, the key thing is that if you have two disjoint closed sets, then their boundaries are disjoint, since they're contained in the sets themselves. Without closedness, two disjoint sets could share boundary points, in which case there's no hope of separating them.

Answer 3

There are lots of different abstract properties defined in maths, some of them even have a name attached. But not all are equally important. For example we do consider spaces were arbitrary intersection of open subsets is open, a.k.a. Alexandrov space. There's a good chance that you will never hear about it, even if you are a professional mathematician. Why? Because it is not important. Why? Because it doesn't occure in problems or applications. It's just rare for a topological space to be Alexandrov.

The same is with those separation axiom. We consider them, because lots of spaces that we study have some or all of them. That's just how the universe is constructed. We only observe those patterns.

As a physicist you should understand this analogy: why the speed of light is finite? It's a similar question. It just is. That's how the universe works.