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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.
95:). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
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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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Topological space where every sequence has a convergent subsequence
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sequential compactness is equivalent to countable compactness.
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262:{\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}}
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380:sequential (Hausdorff) space
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513:Counterexamples in Topology
440:Steen and Seebach, Example
395:Bolzano–Weierstrass theorem
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617:Compactness (mathematics)
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215:first uncountable ordinal
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170:{\displaystyle s_{n}=n}
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99:Examples and properties
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407:Sequence covering maps
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347:{\displaystyle X}
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217:with the
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483:Topology
481:(1999).
389:See also
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