Separation axiom

For the axiom of set theory, see Axiom schema of separation.
Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T3½(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
 Hausdorff)
T6 (perfectly normal
 Hausdorff)
Illustrations of the properties of Hausdorffness, regularity and normality
An illustration of some of the separation axioms. Grey amorphous broken-outline regions indicate open sets surrounding disjoint closed sets or points: red solid-outline circles denote closed sets while black dots represent points.

In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.

The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties.

The precise definitions of the separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition.