# KC-space

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
This is a variation of Hausdorffness. View other variations of Hausdorffness

## Definition

### Symbol-free definition

A topological space is termed a KC-space if every compact subset of it is closed (here, by compact subset, we mean a subset which is a compact space under the subspace topology).

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Hausdorff space any two distinct points can be separated by disjoint open subsets Hausdorff implies KC KC not implies Hausdorff Weakly Hausdorff space|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
weakly Hausdorff space any subset of it arising as the continuous image of a compact Hausdorff space is closed in it |FULL LIST, MORE INFO
US-space KC implies US US not implies KC |FULL LIST, MORE INFO
T1 space points are closed KC implies T1 T1 not implies KC |FULL LIST, MORE INFO
Kolmogorov space any two points can be distinguished (via T1) (via T1) |FULL LIST, MORE INFO