Topology MOC

Compact space

A topological space is called compact every open cover of contains a finite subcover. topology A subset is said to be compact iff it is such under the Subspace topology. A non-compact space may be made compact via a Compactification. A related notion is sequential compactness, which is equivalent in a second-countable space.

Complement characterisation

A topological space is compact iff every family of closed subsets such that has a finite subfamily such that .

Proof

Assume is compact. Let be a family of closed sets with . Then is an open cover of and therefore has a finite subcover , in which case .

For the converse, assume every family of closed subsets of such that has a finite subfamily such that . Let be an open cover of . Then so there exists a finite subfamily such that , in which case is a finite subcover.

Properties

Other useful properties are limited to the Hausdorff-compact space.

• Compactness is a stronger condition than Lindelöf
• The continuous image of a compact space is compact
• Compact subsets of a Hausdorff space are closed
• Closed subsets of a compact space are compact
• A continuous bijection from compact to Hausdorff is a homeomorphism

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