Sequential Compactness

Sequential Compactness 

(in mathematics), a property of spaces. A topological space X is called sequentially compact if every infinite sequence from X has a convergent subsequence. A subset K of X is called sequentially compact if it is sequentially compact as a subspace of X. K is said to be relatively sequentially compact if its closure K is sequentially compact. Related but more important concepts are the concepts of countable compactness and compactness. In a topological space with countable basis as well as in a metric space, sequential compactness, countable compactness, and compactness are equivalent.

A theorem of Weierstrass asserts that every bounded set of real numbers is relatively compact (in the usual metric of the reals). Sets of functions that are compact in one sense or another play a fundamental role in the theory of functions and in functional analysis. An important compactness criterion is given by Arzelà’s theorem: A closed subset of the metric space C of continuous functions on, say, the closed interval [0,1] is compact if and only if it is bounded and equicontinuous.

The relatively compact subsets of Eⁿ (for any n) are precisely its bounded subsets, and its compact subsets are precisely its closed and bounded subsets. In Hilbert space, boundedness does not guarantee relative compactness; a sphere is closed and bounded and yet not compact. An important example of a compact set in Hilbert space is the so-called Hilbert cube consisting of the points with coordinates X_n ≤ 1/2ⁿ. Every compact metric space is homeomorphic to a closed subset of the Hilbert cube (and in the class of topological spaces only the sequentially compact spaces have this property). A subspace L of a normed vector space is finite-dimensional if and only if every closed and bounded subset of L is compact.

REFERENCES

Aleksandrov, P. S. Vvedenie v obshchuiu teoriiu mnozhestv i funktsii. Moscow-Leningrad, 1948.
Hausdorff, F. Teoriia mnozhestv. Moscow-Leningrad, 1937. (Translated from German.)