Understanding separation Axioms in Topology

Question

Weak separation Axioms $(T_0\text{ or Kolmogorov},T_1\text{ or Frechet},T_2\text{ or Hausdorff})$ codify how a topology can distinguish between points in the underlying set, or in the other words how well it can separate points with open sets. They are really intuitive as they show how sequence convergence.

But I couldn't see how Strong separation Axioms $(T_3\text{ or regular }T_1,T_4\text{ or normal }T_1)$ are stronger than Weak separation Axioms. I failed to make an intuition like weak separation Axioms does with sequence convergence.

I see there are different definition for $T_3$ and $T_4$, hence mentioning them below what I am following now:

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• A topology space $(X,\tau)$ is said to be $T_1$ if for any pair of distinct points $x,y\in X$, there exist open sets $U$ and $V$ such that $U$ contain $x$ but not $y$, and $V$ contain $y$ but not $x$.

• A topology space $(X,\tau)$ is said to be regular if for any $x\in X$ and any closed set $C$ not containing $x$, there are disjoint open sets $U$ and $V$ such that $x\in U$ and $C\subseteq V$.

• A topology space $(X,\tau)$ is said to be $T_3$ if it is both $T_1$ and regular.

• A topology space $(X,\tau)$ is said to be normal if for any two disjoint, nonempty, closed subsets $C,D\subseteq X$, there are disjoint open sets $U$ and $V$ containing $C$ and $D$, respectively.

• A topology space $(X,\tau)$ is said to be $T_4$ if it is both $T_1$ and normal.

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I read all of those in fact that course by my own, maybe that's why I couldn't connect them. It will be great help if someone help me to understand/build an intuition on them.

Answer 1

I think one key point is to keep in mind how $T_1$ links points and closed sets: the $T_1$ axiom is equivalent to "all singletons are closed." With this in mind, we can think of strong separation axioms as being the "set-ified" versions of weak separation axioms - if we assume $T_1$-ness as well. For example, normality basically is Hausdorffness if we replace "(distinct) points" with "(disjoint) closed sets," and regularity is a sort of halfway-point between Hausdorffness and normality (we only "set-ify" halfway).

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For example, to see that $T_4$ implies $T_2$ (= Hausdorff), given distinct points $x,y$ in a $T_1$ normal space (= $T_4$ space), by $T_1$-ness we know that $\{x\}$ and $\{y\}$ are disjoint nonempty closed sets so by normality we can find disjoint open $U\ni x, V\ni y$ - and this gives the Hausdorff condition.

The proof that $T_3$ implies $T_2$ is basically the same, except we only "set-ify" one of the points: given distinct points $x,y$ in a $T_1$ space regular space, by $T_1$-ness we get that $\{y\}$ is a nonempty closed set not containing $x$ so by regularity we get disjoint opens $U\ni x, V\ni y$ (the latter since $V\supseteq \{y\}$ is equivalent to $V\ni y$).

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Of course this analogy breaks down if we don't assume $T_1$. For example, the Sierpinski space is normal but neither $T_1$ nor regular.