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The interval C = (2, 4) is not compact because it is not closed (but bounded). The interval B = [0, 1] is compact because it is both closed and bounded. In mathematics , specifically general topology , compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space . [ 1 ]
Since U does not contain any point of K, the set K is already covered by ′ = ′ {}, that is a finite subcollection of the original collection C K. It is thus possible to extract from any open cover C K of K a finite subcover. If a set is closed and bounded, then it is compact. If a set S in R n is bounded, then it can be enclosed within an n-box
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).
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
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 metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
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 ...
Then in , the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak* topology. If X is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded. [1]