# Definition:Kolmogorov Space/Definition 1

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a Kolmogorov space or $T_0$ space if and only if:

$\forall x, y \in S$ such that $x \ne y$, either:
$\exists U \in \tau: x \in U, y \notin U$
or:
$\exists U \in \tau: y \in U, x \notin U$

That is:

for any two distinct elements $x, y \in S$ there exists an open set $U \in \tau$ which contains one of the elements, but not the other.

That is:

$\struct {S, \tau}$ is a $T_0$ space if and only if every two elements in $S$ are topologically distinguishable.

## Also see

• Results about $T_0$ spaces can be found here.

## Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.