Sequentially compact

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.

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. 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 ${\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}}$ copies of the closed unit interval).

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