KC-space
From Topospaces
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
Contents
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 |
