Definition:Kolmogorov Space

Definition

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

Definition 1

$\struct {S, \tau}$ 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:

$\exists U \in \tau: y \in U, x \notin U$

Definition 2

$\struct {S, \tau}$ is a Kolmogorov space or $T_0$ space if and only if no two points can be limit points of each other.

Also see

• Equivalence of Definitions of $T_0$ Space
• Definition:Tychonoff Separation Axioms

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

Source of Name

This entry was named for Andrey Nikolaevich Kolmogorov.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): T-axioms or Tychonoff conditions: 0. ($T_0$-space or Kolmogorov Space)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Kolmogorov space