Supriya Ghosh (Editor)

Countably compact space

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In mathematics a topological space is countably compact if every countable open cover has a finite subcover.

Contents

Examples

  • The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.

    • Properties

  • A compact space is countably compact.
  • A countably compact space is always limit point compact.
  • For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent.
  • The example of the set of all real numbers with the standard topology shows that neither local compactness nor σ-compactness nor paracompactness imply countable compactness.
  • For T1 spaces, countable compactness and limit point compactness are equivalent.

    • References

    Countably compact space Wikipedia
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