Difference between revisions of "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 | + | {{TEX|done}} |
| + | 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==== | ====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 | + | 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. | The product of a countable number of sequentially-compact spaces is sequentially compact. | ||
Latest revision as of 16:41, 22 June 2014
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). If, in addition, the space satisfies the first axiom of countability (cf. also 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, 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
| [a1] | J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4) |
| [a2] | J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 145 |
Sequentially-compact space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sequentially-compact_space&oldid=16065