Equivalent statements to compactness of a metric space/Statement
From Maths
< Equivalent statements to compactness of a metric space
Revision as of 12:41, 1 December 2015 by Alec (Talk | contribs) (Created page with "<noinclude> ==Statement of theorem== </noinclude> Given a metric space {{M|(X,d)}}, the following are equivalent{{rITTGG}}<ref group="Note">To say statements are equivalen...")
Statement of theorem
Given a metric space (X,d), the following are equivalent[1][Note 1]:
- X is compact
- Every sequence in X has a subsequence that converges (has a convergent subsequence)
- X is totally bounded and complete
Notes
- Jump up ↑ To say statements are equivalent means we have one ⟺ one of the other(s)
References
