Compact space

Hausdorff-compact space

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

A topological space will be called Hausdorff-compact if it is both Hausdorff and 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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