Sequentially compact space

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In mathematics, a topological space is sequentially compact if every infinite sequence has a convergent subsequence. For general topological spaces, the notions of compactness and sequential compactness are not equivalent; they are, however, equivalent for metric spaces.

Examples and properties[edit]

The space of all real numbers with the standard topology is not sequentially compact; the sequence (sn = n) for all natural numbers n is a sequence which has no convergent subsequence.

If a space is a metric space, then it is sequentially compact if and only if it is compact.[1] 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 of 2^{\aleph_0}=\mathfrak c copies of the closed unit interval).[2]

Related notions[edit]

In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are equivalent.

In a sequential space sequential compactness is equivalent to countable compactness.[3]

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 [4]

See also[edit]

Notes[edit]

  1. ^ Willard, 17G, p. 125.
  2. ^ Steen and Seebach, Example 105, pp. 125—126.
  3. ^ Engelking, General Topology, Theorem 3.10.31
    K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d3 (by P. Simon)
  4. ^ Brown, Ronald, "Sequentially proper maps and a sequential compactification", J. London Math Soc. (2) 7 (1973) 515-522.

References[edit]