Sequentially compact space

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. A metric space X is sequentially compact if every sequence has a convergent subsequence which converges to a point in X.

Examples and properties

The space of all real numbers with the standard topology is not sequentially compact; the sequence (s_n = n) for all natural numbers n is a sequence that 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 that are not compact (such as the first uncountable ordinal with the order topology), and compact spaces that are not sequentially compact (such as the product of $2^{\aleph _{0}}={\mathfrak {c}}$ copies of the closed unit interval).^[2]

Related notions

• A topological space X is said to be limit point compact if every infinite subset of X has a limit point in X.
• A topological space is countably compact if every countable open cover has a finite subcover.

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

In a sequential (Hausdorff) 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

• Bolzano–Weierstrass theorem

Notes

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

• Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). ISBN 0-03-079485-4.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.