The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts-until it closes down on the desired limit point. On the one hand, Bernard Bolzano ( 1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point.īolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. 3.3.4 Characterization by continuous functions.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. 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. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets.
The most useful notion-and 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. In general topological spaces, however, these notions of compactness are not necessarily equivalent. Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces. Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Similarly, the space of rational numbers Q, the entire real number line, the sequence of points 0, 1, 2, 3, ., has no subsequence that converges to any real number.Ĭompactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. For example, the "unclosed" interval (0,1) would not be compact because it excludes the "limiting values" of 0 and 1, whereas the closed interval would be compact. that the space not exclude any "limiting values" of points. In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "holes" or "missing endpoints", i.e. The interval B = is compact because it is both closed and bounded. The interval C = (2, 4) is not compact because it is not closed. Per the compactness criteria for Euclidean space as stated in the Heine–Borel theorem, the interval A = (−∞, −2] is not compact because it is not bounded.
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