Compactness axioms

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

We say that a topological space X is

  • Compact if every cover by open sets has a finite subcover.
  • 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.

References