Kolmogorov space

Definition

A topological space $X$ is termed a $T_{0}$ space or Kolmogorov space if it satisfies the following equivalent conditions:

1. For any two distinct points $x,y\in X$, there is either an open subset containing $x$ but not $y$, or an open subset containing $y$ but not $x$.
2. For any two distinct points $x,y\in X$, there is either a closed subset containing $x$ but not $y$, or a closed subset containing $y$ but not $x$.
3. For any two distinct points $x,y\in X$ such that ${\overline {\{x\}}}\neq {\overline {\{y\}}}$, i.e., the closures of any two distinct one-point subsets must be distinct.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T0

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
T1 space	singleton subsets are closed		Kolmogorov not implies T1
Hausdorff space				KC-space, Locally Hausdorff space, Sober T0 space|FULL LIST, MORE INFO
locally Hausdorff space				|FULL LIST, MORE INFO
totally disconnected space				|FULL LIST, MORE INFO
regular space				|FULL LIST, MORE INFO
normal space				|FULL LIST, MORE INFO
metrizable space				Functionally Hausdorff space, Hausdorff space, Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO