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: $T_0$
We say that a topological space $(X, \vartheta)$ is $T_0$ when for any two points $x,y \in X$ there exists an open set $U \in \vartheta$ which contains one of the points, but not the other.

Said otherwise, $(X, \vartheta)$ is $T_0$ when for any two points $x,y \in X$ at least one of the following options happens:
 * There exists $U \in \vartheta$ such that $x \in U$ and $y \notin U$.
 * There exists $U \in \vartheta$ such that $y \in U$ and $x \notin U$.

Yet another way to phrase this is to say that $(X,\vartheta)$ is $T_0$ if every two points in $X$ are topologically distinguishable.

Such a space is called a Kolmogorov space.

Definition: $T_1$
We say that a topological space $(X, \vartheta)$ is $T_1$ when for any two points $x,y \in X$ there exist open sets $U,V \in \vartheta$ such that $x$ is in $U$ but not in $V$, and $y$ is in $V$ but not in $U$; this is, if both of the following happen:
 * $x \in U$ and $y \notin U$.
 * $y \in V$ and $x \notin V$.

That is: $(X,\vartheta)$ is $T_1$ when every two points in $X$ are separated.

Definition: $T_2$
We say that a topological space $(X, \vartheta)$ is $T_2$ when for any two points $x,y \in X$ there exist open sets $U,V \in \vartheta$ such that $x \in U$, $y \in V$ and $U \cap V = \emptyset$.

That is: $(X,\vartheta)$ is $T_2$ when every two points in $X$ are separated by neighborhoods.

This condition is also known as the Hausdorff condition, and such a topological space $(X, \vartheta)$ is known as a Hausdorff space.

Conveniently, a topological space is Hausdorff if any two distinct points can be housed off from one another in separate disjoint open sets.

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.