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:

$\forall x, y \in X$ such that $x \ne y$, 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 distinct 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.

Source of Name

This entry was named for Andrey Kolmogorov.

Also see

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

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$