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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.
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Brown, Ronald, "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.
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is an example of a sequentially compact topological space that is not compact. The
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is naturally a topological space, and for metric spaces, the notions of
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is an example of a compact space that is not sequentially compact.
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Topological space where every sequence has a convergent subsequence
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sequential compactness is equivalent to countable compactness.
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and sequential compactness are equivalent (if one assumes
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262:{\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}}
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622:Properties of topological spaces
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380:sequential (Hausdorff) space
363:has a finite subcover. In a
307:if every infinite subset of
513:Counterexamples in Topology
440:Steen and Seebach, Example
395:Bolzano–Weierstrass theorem
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617:Compactness (mathematics)
528:Willard, Stephen (2004).
215:first uncountable ordinal
60:converging to a point in
170:{\displaystyle s_{n}=n}
137:{\displaystyle (s_{n})}
99:Examples and properties
532:. Dover Publications.
508:Seebach, J. Arthur Jr.
407:Sequence covering maps
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347:{\displaystyle X}
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361:open cover
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239:ℵ
217:with the
144:given by
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483:Topology
481:(1999).
389:See also
177:for all
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223:product
211:compact
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