Definition:Tychonoff Separation Axioms

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 by 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.