Definition:Tychonoff Separation Axioms

Definition
The Separation Axioms (sometimes known as the Kolmogorov Separation Axioms or the Tychonoff Separation Axioms) are a classification system for topological spaces such that each condition is stronger than the predecessor; that is to say, a $T_2$ space is necessarily $T_1$ as well, but there exist $T_1$ spaces which are not $T_2$.

Apart from the ones below, other kinds of $T$ spaces have been defined, but these definitions vary from author to author.

Definition: Regular
We say that a topological space $(X, \vartheta)$ is regular when for any closed set $F \subseteq X$ and any point $x \in X$ such that $x \notin F$ there exist open sets $U, V \in \vartheta$ such that $F \subseteq U$, $y \in V$ and $U \cap V = \emptyset$.

That is: $(X,\vartheta)$ is regular when any closed subset $F \subseteq X$ and any point not in $F$ are separated by neighborhoods.

Definition: Normal
We say that a topological space $(X, \vartheta)$ is normal when for any two disjoint closed sets $E, F \subseteq X$ there exist open sets $U, V \in \vartheta$ such that $E \subseteq U$, $F \subseteq V$ and $U \cap V = \emptyset$.

That is: $(X,\vartheta)$ is normal when any two disjoint closed subsets of $X$ are separated by neighborhoods.