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• Compact_space abstract "In the mathematical discipline of general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, all points within some fixed distance of each other). This notion is generalized to more general topological spaces in various ways. For instance, a space is sequentially compact if any infinite sequence of points sampled from the space must eventually, infinitely often, get arbitrarily close to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sense if and only if it is closed and bounded. Examples include a closed interval or a rectangle. Thus if one chooses an infinite number of points in the closed unit interval, some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … get arbitrarily close to 0. (Also, some get arbitrarily close to 1.) The same set of points would not have, as a limit point, any point of the open unit interval; 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, … has no sub-sequence that ultimately gets arbitrarily close to any given real number.Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions. The term compact was introduced into mathematics by Maurice Fréchet in 1906 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation. A typical application is furnished by the Arzelà–Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that "cover" the space in the sense that each point of the space must lie in some set contained in the family. The standard unqualified use of the term compact in mathematics usually means compactness in this latter sense. This more subtle definition 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.".
• Compact_space thumbnail Compact.svg?width=300.
• Compact_space wikiPageExternalLink v=onepage&q=.
• Compact_space wikiPageExternalLink v=onepage&q=.
• Compact_space wikiPageExternalLink item?id=ASENS_1895_3_12__9_0.
• Compact_space wikiPageID "6042".
• Compact_space wikiPageRevisionID "605317954".
• Compact_space first "A.V.".
• Compact_space hasPhotoCollection Compact_space.
• Compact_space id "1233".
• Compact_space id "3133".
• Compact_space id "C/c023530".
• Compact_space last "Arkhangel'skii".
• Compact_space title "Compact space".
• Compact_space title "Countably compact".
• Compact_space title "Examples of compact spaces".
• Compact_space subject Category:Compactness_(mathematics).
• Compact_space subject Category:General_topology.
• Compact_space subject Category:Properties_of_topological_spaces.
• Compact_space subject Category:Topology.
• Compact_space type Abstraction100002137.
• Compact_space type Possession100032613.
• Compact_space type PropertiesOfTopologicalSpaces.
• Compact_space type Property113244109.
• Compact_space type Relation100031921.
• Compact_space comment "In the mathematical discipline of general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, all points within some fixed distance of each other). This notion is generalized to more general topological spaces in various ways.".
• Compact_space label "Compacité (mathématiques)".
• Compact_space label "Compact space".
• Compact_space label "Compacte ruimte".
• Compact_space label "Espacio compacto".
• Compact_space label "Espaço compacto".
• Compact_space label "Kompakter Raum".
• Compact_space label "Przestrzeń zwarta".
• Compact_space label "Spazio compatto".
• Compact_space label "Компактное пространство".
• Compact_space label "فضاء متراص".
• Compact_space label "コンパクト空間".
• Compact_space label "紧空间".
• Compact_space sameAs Kompaktní_množina.
• Compact_space sameAs Kompakter_Raum.
• Compact_space sameAs Συμπαγής_χώρος.
• Compact_space sameAs Espacio_compacto.
• Compact_space sameAs Espazio_trinko.
• Compact_space sameAs Compacité_(mathématiques).
• Compact_space sameAs Spazio_compatto.
• Compact_space sameAs コンパクト空間.
• Compact_space sameAs 콤팩트_공간.
• Compact_space sameAs Compacte_ruimte.
• Compact_space sameAs Przestrzeń_zwarta.
• Compact_space sameAs Espaço_compacto.
• Compact_space sameAs m.01tcj.
• Compact_space sameAs Mx4rwScB7JwpEbGdrcN5Y29ycA.
• Compact_space sameAs Q381892.
• Compact_space sameAs Q381892.
• Compact_space sameAs Compact_space.
• Compact_space wasDerivedFrom Compact_space?oldid=605317954.
• Compact_space depiction Compact.svg.
• Compact_space isPrimaryTopicOf Compact_space.