I was looking around to understand better the separation axioms, when I found this sentence in the wolfram page concerning the separation axioms.
[talking about the the separation axioms...]
...each of them tells us how tightly a closed subset can be wrapped in an open set. The measure of tightness is the extent to which this envelope can separate the subset from other subsets.
This looks very interesting but I don't really understand it. What does it mean when applied to T0, T1, T2, T3, T4?
$T_1$: A closed set that contains just one point can be wrapped in an open set that separates it from any one other prescribed point.
$T_2$: A closed set that contains just one point can be wrapped in an open set that separates it from a whole little neighborhood of any one other prescribed point.
$T_3$: Any closed set $C$ (not necessarily just a single point) can be wrapped in an open set that separates it from some neighborhood of any one other prescribed point not in $C$. Equivalently, any closed set that contains just one point $p$ can be wrapped in an open set that separates it from some neighborhood of any closed set that doesn't contain $p$.
$T_4$: Any closed set $C$ (not necessarily just a single point) can be wrapped in an open set that separates it from some neighborhood of any closed set disjoint from $C$.