View source for Sequentially-compact space

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A topological space in which any infinite sequence of points contains a convergent subsequence (the Bolzano–Weierstrass condition). In the class of $T_1$-spaces, a sequentially-compact space is countably compact (see [[Compactness, countable|Compactness, countable]]). If, in addition, the space satisfies the [[First axiom of countability|first axiom of countability]] (cf. also [[Base|Base]]), then its countable compactness implies that it is sequentially compact. A sequentially-compact space need not be compact; for example, the set of all ordinal numbers less than the first uncountable number, equipped with the topology whose base is the set of all open intervals.



====Comments====
Some authors take the Bolzano–Weierstrass condition to mean that every infinite set has an [[Accumulation point|accumulation point]], which in the class of $T_1$-spaces is equivalent to countable compactness.

The product of a countable number of sequentially-compact spaces is sequentially compact.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,   "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , v. Nostrand  (1955)  pp. 145</TD></TR></table>