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The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.
Note that these measures of non-compactness are useless for subsets of Euclidean space R n: by the Heine–Borel theorem, every bounded closed set is compact there, which means that γ(X) = 0 or ∞ according to whether X is bounded or not. Measures of non-compactness are however useful in the study of infinite-dimensional Banach spaces, for ...
Lemma: A closed subset of a compact set is compact. Let K be a closed subset of a compact set T in R n and let C K be an open cover of K. Then U = R n \ K is an open set and = {} is an open cover of T. Since T is compact, then C T has a finite subcover ′, that also covers the smaller set K.
A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded. [2] Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded).
Functions with compact support on a topological space are those whose closed support is a compact subset of . If X {\displaystyle X} is the real line, or n {\displaystyle n} -dimensional Euclidean space, then a function has compact support if and only if it has bounded support , since a subset of R n {\displaystyle \mathbb {R} ^{n}} is compact ...
The Heine–Borel theorem asserts that a subset of the real line is compact if and only if it is both closed and bounded. Correspondingly, a metric space has the Heine–Borel property if every closed and bounded set is also compact. The concept of a continuous function can likewise be generalized.
However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact. As a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded. The norm is (sequentially) weakly lower-semicontinuous: if converges weakly to x, then
Every compact metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace of R n is compact and therefore complete. [1]