What are the nonempty compact sets of the Euclidean topology on the set of reals ?
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One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.[3] The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0 (while others accumulate to 1). The same set of points would not accumulate to any point of the open unit interval (0, 1); so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points 0, 1, 2, 3, …, which is not bounded, has no subsequence that converges to any real number.
Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces are encountered in mathematical analysis, where the property of compactness of some topological spaces arises in the hypotheses or in the conclusions of many fundamental theorems, such as the Bolzano–Weierstrass theorem, the extreme value theorem, the Arzelà–Ascoli theorem, and the Peano existence theorem. Another example is the definition of distributions, which uses the space of smooth functions that are zero outside of some (unspecified) compact space.
Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces.[4] In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness, is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.
The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well.
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