Sequentially compact space

In mathematics, a topological space is sequentially compact if every 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

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 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 uncountably many 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 space sequential compactness is equivalent to countable compactness.^[3]

See also

• Bolzano–Weierstrass theorem

Notes

References

• Munkres, James (1999). Topology (2nd edition 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 0030794854.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.