Compactness is a property of a topological space that makes it behave in many ways like a finite set, for instance so that every continuous real-valued function on a compact space is bounded and attains its maximum and minimum, generalizing the extreme value theorem. A topological space is compact if every open cover of it has a finite subcover, and for subsets of Euclidean space this is equivalent, by the Heine-Borel theorem, to being closed and bounded. Compactness was formally introduced by Maurice Frechet in 1906, generalizing the Bolzano-Weierstrass theorem, and Pavel Alexandrov and Pavel Urysohn later developed the open-cover formulation now standard in topology. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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Metric Space, Concepts Compactness was first formulated for spaces of functions with a distance notion (Frechet, 1906) and in a metric space it coincides with the space being both complete and totally bounded, the classical form of the concept.
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compactness is a property of a space that makes it behave in many ways like a finite set
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Compactness was formally introduced by Maurice Fréchet in 1906 in work generalizing the Bolzano-Weierstrass theorem from sets of points to spaces of functions
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