[[Compact space]]
# Hausdorff-compact space

The notion of [[Compact space|compactness]] becomes especially useful when the space is also [[Hausdorff space|Hausdorff]].
The below definition is only tagged as such so that proof graphs display correctly.

> A topological space $X$ will be called **Hausdorff-compact** if it is both [[Hausdorff space|Hausdorff]] and [[Compact space|compact]]. #m/def/topology 

## Properties

- Subsets are closed iff they are compact, which follows from:
  - [[Closed subsets of a compact space are compact]]
  - [[Compact subsets of a Hausdorff space are closed]]
- [[No finer compact, no coarser Hausdorff than a Hausdorff-compact]]
- [[Hausdorff-compact implies normal]]

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