Equivalent statements to compactness of a metric space/Statement

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Statement of theorem

Given a metric space [ilmath](X,d)[/ilmath], the following are equivalent[1][Note 1]:

  1. [ilmath]X[/ilmath] is compact
  2. Every sequence in [ilmath]X[/ilmath] has a subsequence that converges (has a convergent subsequence)
  3. [ilmath]X[/ilmath] is totally bounded and complete


Notes

  1. ↑ To say statements are equivalent means we have one [ilmath]\iff[/ilmath] one of the other(s)

References

  1. ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene



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