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| In [[mathematics]], a [[topological space]] is '''sequentially compact''' if every [[infinite sequence]] has a [[limit of a sequence|convergent]] [[subsequence]]. For general topological spaces, the notions of compactness and sequential [[compactness]] are not equivalent; they are, however, equivalent for [[metric spaces]].
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| == Examples and properties ==
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| The space of all [[real number]]s with the [[standard topology]] is not sequentially compact; the sequence {{nowrap|(''s<sub>n</sub>'' {{=}} ''n'')}} for all [[natural number]]s ''n'' is a sequence which has no convergent subsequence.
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| If a space is a [[metric space]], then it is sequentially compact if and only if it is [[Compact space|compact]].<ref>Willard, 17G, p. 125.</ref> However in general there exist sequentially compact spaces which are not compact (such as the [[first uncountable ordinal]] with the [[order topology]]), and compact spaces which are not sequentially compact (such as the [[product topology|product]] of <math>2^{\aleph_0}=\mathfrak c</math> copies of the [[closed unit interval]]).<ref>Steen and Seebach, Example '''105''', pp. 125—126.</ref>
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| == Related notions ==
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| *A topological space ''X'' is said to be [[limit point compact]] if every infinite subset of ''X'' has a [[limit point]] in ''X''.
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| *A topological space is [[countably compact space|countably compact]] if every countable [[open cover]] has a finite subcover.
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| In a [[metric space]], the notions of sequential compactness, limit point compactness, countable compactness and [[compact space|compactness]] are equivalent.
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| In a [[sequential space]] sequential compactness is equivalent to countable compactness.<ref>Engelking, General Topology, Theorem 3.10.31<br> K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d3 (by P. Simon)
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| </ref>
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| There is also a notion of a one-point sequential compactification -- the idea is that the non convergent sequences should all converge to the extra point. See
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| <ref>Brown, Ronald, "Sequentially proper maps and a sequential
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| compactification", J. London Math Soc. (2) 7 (1973)
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| 515-522.
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| </ref>
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| == See also ==
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| *[[Bolzano–Weierstrass theorem]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book
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| | author = [[James Munkres|Munkres, James]]
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| | year = 1999
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| | title = Topology
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| | edition = 2nd edition
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| | publisher = [[Prentice Hall]]
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| | isbn = 0-13-181629-2
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| }}
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| * [[Lynn Arthur Steen|Steen, Lynn A.]] and [[J. Arthur Seebach, Jr.|Seebach, J. Arthur Jr.]]; ''[[Counterexamples in Topology]]'', Holt, Rinehart and Winston (1970). ISBN 0-03-079485-4.
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| *{{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | id=ISBN 0-486-43479-6}}
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| {{topology-stub}}
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| [[Category:Compactness (mathematics)]]
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Nice to satisfy you, my title is Refugia. His family life in South Dakota but his spouse wants them to transfer. To do aerobics is a factor that I'm totally addicted to. Supervising is my occupation.
Feel free to surf to my page: streaming.iwarrior.net