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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==Statement of theorem== | ==Statement of theorem== | ||
</noinclude> | </noinclude> | ||
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]] ( | + | # Every [[sequence]] in {{M|X}} has a [[subsequence]] that [[convergent (sequence)|converges]] ({{AKA}}: having a ''convergent subsequence'') |
| − | # {{M|X}} is [[totally bounded]] and [[complete metric space|complete]] | + | # {{M|X}} is [[totally bounded]] and [[complete metric space|complete]]<!-- |
| − | + | --><noinclude> | |
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| − | <noinclude> | + | |
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 11:39, 27 May 2016
- Attention people coming in from search engines: this is a sub-page, you want Equivalent statements to compactness of a metric space
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
