Difference between revisions of "Equivalent statements to compactness of a metric space/Statement"
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(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...") |
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Given a [[metric space]] {{M|(X,d)}}, the following are equivalent{{rITTGG}}<ref group="Note">To say statements are equivalent means we have one {{M|\iff}} one of the other(s)</ref>: | Given a [[metric space]] {{M|(X,d)}}, the following are equivalent{{rITTGG}}<ref group="Note">To say statements are equivalent means we have one {{M|\iff}} one of the other(s)</ref>: | ||
# {{M|X}} is [[compact]] | # {{M|X}} is [[compact]] | ||
| − | # Every [[sequence]] in {{M|X}} has a [[subsequence]] that [[converges]] (has a ''convergent subsequence'') | + | # Every [[sequence]] in {{M|X}} has a [[subsequence]] that [[convergent (sequence)|converges]] (has a ''convergent subsequence'') |
# {{M|X}} is [[totally bounded]] and [[complete metric space|complete]] | # {{M|X}} is [[totally bounded]] and [[complete metric space|complete]] | ||
Revision as of 16:15, 1 December 2015
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 (has 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
