I'm working through Topology Through Inquiry, and while I grok the meaning of T1 and T2 spaces (they roughly quantify the ability of a topology to separate the elements in the space), what the meaning of T3 and T4 spaces is less clear.
I'm not really asking for a definition (that's plenty clear to me), but rather an intuition for why being able to "separate" closed sets makes a topology "more able to separable" beyond "points are the smallest possible closed sets".
Maybe it helps you to see this the particular case of compact spaces. Compact topological spaces in $\mathbb{R}^n$ have this notion of being "smallish" (bounded) and closed. This notion is difficult to translate to general compact topological spaces, but when you have a compact $T_2$ space, there is a result telling you that it will also be a $T_4$ space. A way a professor of mine put this is that the closed sets of compact spaces are "small enough" to pass as points, recovering the sense of a compact space being something related to smallness. In this sense, spaces that are $T_2$ but not $T_4$ can be thought of as spaces which have points that can easily slip into small places without bothering each others, but whose closed sets are too large in general for this (even though you don't have a notion of size). And I like to describe $T_4$ as spaces that not only allow points to easily slip into places, but also have closed sets slim enough that they can pass as points
Aside from the literal definition itself, the intuition I have in my head for $T_4$ spaces is that they're ones with enough continuous functions on them for the Tietze extension theorem to hold (and that theorem is equivalent to $T_4$-ness, at least in the Hausdorff case). The main difficulty in the proof of the Tietze extension theorem (or, equivalently, Urysohn's lemma) is just getting any nonconstant continuous functions $X \to [0, 1]$ at all; there's no obvious way of constructing them on an arbitrary topological space $X$, and you have to work a bit to show that the normality of $X$ gives a filtration of $X$ by sets indexed in $\mathbb{Z}[\frac{1}{2}] \cap [0, 1]$ that's fine enough to induce a continuous map $X \to [0, 1]$. For metric spaces, the metric itself gives the required extension. It's not true in general that $T_4$ spaces are metrizable (e.g., the Sorgenfrey line), but it's a weakening of that property that ensures enough continuous functions to do analysis with.
The unfair intuition I have in my head for $T_4$ spaces is that they're left over from an earlier period in topology where one of the major active goals was to find criteria for a space be metrizable. That question has largely been answered by results like the Nagata-Smirnov, Bing, and Urysohn metrization theorems, and I'm not sure it's still of interest to anyone not working directly in topology (but see the earlier remark about unfairness here). So in that sense, $T_4$-ness is an approximation to metrizability that still gives you some interesting topology but allows for more pathology than more familiar spaces like manifolds, CW-complexes, etc.
