Definition:Kolmogorov Space

Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.

$\left({X, \vartheta}\right)$ is a Kolmogorov space or $T_0$ space iff:


 * $\forall x, y \in X$, either:
 * $\exists U \in \vartheta: x \in U, y \notin U$
 * or:
 * $\exists U \in \vartheta: y \in U, x \notin U$

That is, 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.

That is:
 * $\left({X, \vartheta}\right)$ is a $T_0$ space iff every two points in $X$ are topologically distinguishable.

Equivalent Definitions
$\left({X, \vartheta}\right)$ is a Kolmogorov space or $T_0$ space iff no two points can be limit points of each other.

This is proved in Equivalent Definitions for $T_0$ Space.