# Kolmogorov space

From Topospaces

## Definition

A topological space is termed a space or **Kolmogorov space** if it satisfies the following equivalent conditions:

- For any two distinct points , there is either an open subset containing but not , or an open subset containing but not .
- For any two distinct points , there is either a closed subset containing but not , or a closed subset containing but not .
- For any two distinct points such that , 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 |