Definition:Tychonoff Separation Axioms

The Kolmogorov 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 $$T_2$$.

For all of these definitions, $$X$$ is taken to be a topological space with topology $$\vartheta$$ and $$x,y \in X$$.

= $$T_0$$ =

$$\exists U \in \vartheta$$ such that either $$x \in U$$ or $$y \in U$$, but not both.

= $$T_1$$ =

$$\exists U, V \in \vartheta$$ such that $$x \in U$$ and $$y \in V$$ where $$y \notin U$$ and $$ x \notin V$$.

= $$T_2$$ =

$$\exists U, V \in \vartheta$$ such that $$x \in U$$ and $$y \in V$$ where $$U \cap V = \varnothing$$.

Other kinds of $$T$$ spaces have been defined, but these definitions vary from author to author.