Compactness axioms

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In general topology, the important property of compactness has a number of related properties.

The definitions require some preliminary terminology. A cover of a set X is a family \mathcal{U} = \{ U_\alpha : \alpha \in A \} such that the union \bigcup_{\alpha \in A} U_\alpha is equal to X. A subcover is a subfamily which is again a cover \mathcal{S} = \{ U_\alpha : \alpha \in B \} where B is a subset of A. A refinement is a cover \mathcal{R} = \{ V_\beta : \beta \in B \} such that for each β in B there is an α in A such that V_\beta \subseteq U_\alpha. A cover is finite or countable if the index set is finite or countable. A cover is point finite if each element of X belongs to a finite numbers of sets in the cover. The phrase "open cover" is often used to denote "cover by open sets".

Definitions

We say that a topological space X is

  • Compact if every cover by open sets has a finite subcover.
  • A compactum if it is a compact metric space.
  • Countably compact if every countable cover by open sets has a finite subcover.
  • Lindelöf if every cover by open sets has a countable subcover.
  • Sequentially compact if every convergent sequence has a convergent subsequence.
  • Paracompact if every cover by open sets has an open locally finite refinement.
  • Metacompact if every cover by open sets has a point finite open refinement.
  • Orthocompact if every cover by open sets has an interior preserving open refinement.
  • σ-compact if it is the union of countably many compact subspaces.
  • Locally compact if every point has a compact neighbourhood.
  • Strongly locally compact if every point has a neighbourhood with compact closure.
  • σ-locally compact if it is both σ-compact and locally compact.
  • Pseudocompact if every continuous real-valued function is bounded.
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