Difference between revisions of "Equivalent statements to compactness of a metric space/Statement"

Revision as of 12:41, 1 December 2015

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