See this video – and this video

• $T_1$ . Frechet space. Every pair of points have neighbourhoods not containing the other point (equivalent to points being closed) – vid
• $T_2$ (Hausdorff space)
• $T_3$. A point and a closed subset not containing the point, then they have disjoint neighbourhoods.
• $T_4$. Disjoint closed subsets have disjoint neighbourhoods.
• ...

https://en.wikipedia.org/wiki/Separation_axiom

Regular space

Normal space

Then normal => regular => Hausdorff => $T_1$

Lemma.

X is T3 if and only if for every $x \in X$ and neighbourhood $U$ of $X$, there is a nbhd $V$ of $x$ s.t. $\bar{V} \subset U$.

X is T4 iff for each closed subsete $A$ of $X$ and nbhd $U$ of $A$, there is a nbhd $V$ of $A$ s.t. $\bar{V} \subset U$

Urysohn's lemma –> Urysohn metrization theorem

Proposition. Every Regular space with a countable basis is normal