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]}:
- [ilmath]X[/ilmath] is compact
- Every sequence in [ilmath]X[/ilmath] has a subsequence that converges (AKA: having a convergent subsequence)
- [ilmath]X[/ilmath] is totally bounded and complete
Notes
- ↑ To say statements are equivalent means we have one [ilmath]\iff[/ilmath] one of the other(s)
References