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
- J.L. Kelley (1955), General topology, van Nostrand
- Steen, Lynn Arthur & J. Arthur Jr. Seebach (1978), Counterexamples in Topology, Berlin, New York: Springer-Verlag