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Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set. Any global field K is a discrete additive subgroup of its adele ring, and the quotient space is compact. This was used in John Tate's thesis to allow harmonic analysis to be used in number theory.
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.
(In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.) Briefly, a closed set contains all of its boundary points , while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number.
If is the real line, or -dimensional Euclidean space, then a function has compact support if and only if it has bounded support, since a subset of is compact if and only if it is closed and bounded. For example, the function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined above is a continuous function with compact support ...
First we prove the theorem for (set of all real numbers), in which case the ordering on can be put to good use. Indeed, we have the following result: Indeed, we have the following result: Lemma : Every infinite sequence ( x n ) {\displaystyle (x_{n})} in R 1 {\displaystyle \mathbb {R} ^{1}} has an infinite monotone subsequence (a subsequence ...
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.
If is a compact topological space, and () is a monotonically increasing sequence (meaning () + for all and ) of continuous real-valued functions on which converges pointwise to a continuous function :, then the convergence is uniform.
But not every open set in a weakly locally compact space is weakly locally compact. For example, the one-point compactification of the rational numbers is compact, and hence weakly locally compact. But it contains as an open set which is not weakly locally compact. Quotient spaces of locally compact Hausdorff spaces are compactly generated ...