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- Compact_space abstract "In mathematics, and more specifically in 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, having all its points lie within some fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.One such generalization is that a space is sequentially compact if any infinite sequence of points sampled from the space must frequently (infinitely often) get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space. The Heine-Borel 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 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, … accumulate to 0 (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, … has no subsequence that converges 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 1904 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 or 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, 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 must lie 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 a synonym for compact space, but usually refers to a compact subspace of a topological space.".
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- Compact_space wikiPageWikiLink Accumulation_point.
- Compact_space wikiPageWikiLink Alaoglus_theorem.
- Compact_space wikiPageWikiLink Alexanders_sub-base_theorem.
- Compact_space wikiPageWikiLink Alexandroff_extension.
- Compact_space wikiPageWikiLink Alexandroff_one-point_compactification.
- Compact_space wikiPageWikiLink Algebraic_geometry.
- Compact_space wikiPageWikiLink Annales_Scientifiques_de_lxc3x89cole_Normale_Supxc3xa9rieure.
- Compact_space wikiPageWikiLink Archimedean_property.
- Compact_space wikiPageWikiLink Arzelà–Ascoli_theorem.
- Compact_space wikiPageWikiLink Axiom_of_choice.
- Compact_space wikiPageWikiLink Banach_algebra.
- Compact_space wikiPageWikiLink Banach_space.
- Compact_space wikiPageWikiLink Banach–Alaoglu_theorem.
- Compact_space wikiPageWikiLink Bernard_Bolzano.
- Compact_space wikiPageWikiLink Bisection_method.
- Compact_space wikiPageWikiLink Bolzano–Weierstrass_theorem.
- Compact_space wikiPageWikiLink Boundary_(topology).
- Compact_space wikiPageWikiLink Bounded_linear_operator.
- Compact_space wikiPageWikiLink Bounded_operator.
- Compact_space wikiPageWikiLink Bounded_set.
- Compact_space wikiPageWikiLink Cantor_set.
- Compact_space wikiPageWikiLink Category:Compactness_(mathematics).
- Compact_space wikiPageWikiLink Category:General_topology.
- Compact_space wikiPageWikiLink Category:Properties_of_topological_spaces.
- Compact_space wikiPageWikiLink Category:Topology.
- Compact_space wikiPageWikiLink Cesare_Arzelà.
- Compact_space wikiPageWikiLink Circolo_Matematico_di_Palermo.
- Compact_space wikiPageWikiLink Closed_interval.
- Compact_space wikiPageWikiLink Closed_set.
- Compact_space wikiPageWikiLink Closed_unit_ball.
- Compact_space wikiPageWikiLink Cocountable_topology.
- Compact_space wikiPageWikiLink Cofinite_topology.
- Compact_space wikiPageWikiLink Cofiniteness.
- Compact_space wikiPageWikiLink Commutative_ring.
- Compact_space wikiPageWikiLink Compact_group.
- Compact_space wikiPageWikiLink Compact_operator.
- Compact_space wikiPageWikiLink Compactification_(mathematics).
- Compact_space wikiPageWikiLink Compactly_generated_space.
- Compact_space wikiPageWikiLink Complete_accumulation_point.
- Compact_space wikiPageWikiLink Complete_lattice.
- Compact_space wikiPageWikiLink Complete_metric_space.
- Compact_space wikiPageWikiLink Completely_regular_space.
- Compact_space wikiPageWikiLink Completeness_(topology).
- Compact_space wikiPageWikiLink Complex_number.
- Compact_space wikiPageWikiLink Continuous_function.
- Compact_space wikiPageWikiLink Continuous_function_(topology).
- Compact_space wikiPageWikiLink Convergence_of_random_variables.
- Compact_space wikiPageWikiLink Counterexamples_in_Topology.
- Compact_space wikiPageWikiLink Cover_(topology).
- Compact_space wikiPageWikiLink David_Hilbert.
- Compact_space wikiPageWikiLink Dense_topological_subspace.
- Compact_space wikiPageWikiLink Discrete_space.
- Compact_space wikiPageWikiLink Discrete_topology.
- Compact_space wikiPageWikiLink Disk_(mathematics).
- Compact_space wikiPageWikiLink Dover_Publications.
- Compact_space wikiPageWikiLink Eberlein_compactum.
- Compact_space wikiPageWikiLink Eduard_Heine.
- Compact_space wikiPageWikiLink Empty_set.
- Compact_space wikiPageWikiLink Erhard_Schmidt.
- Compact_space wikiPageWikiLink Euclidean_space.
- Compact_space wikiPageWikiLink Exhaustion_by_compact_sets.
- Compact_space wikiPageWikiLink Extreme_value_theorem.
- Compact_space wikiPageWikiLink Filter_(mathematics).
- Compact_space wikiPageWikiLink Finite_intersection_property.
- Compact_space wikiPageWikiLink Finite_set.
- Compact_space wikiPageWikiLink Finite_topological_space.
- Compact_space wikiPageWikiLink Finite_topology.
- Compact_space wikiPageWikiLink First-countable.
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- Compact_space wikiPageWikiLink First-order_logic.
- Compact_space wikiPageWikiLink First_isomorphism_theorem.
- Compact_space wikiPageWikiLink Function_space.
- Compact_space wikiPageWikiLink Function_spaces.
- Compact_space wikiPageWikiLink General_linear_group.
- Compact_space wikiPageWikiLink General_topology.
- Compact_space wikiPageWikiLink Giulio_Ascoli.
- Compact_space wikiPageWikiLink Greens_function.
- Compact_space wikiPageWikiLink Hausdorff_space.
- Compact_space wikiPageWikiLink Hausdorff_spaces.
- Compact_space wikiPageWikiLink Heine-Borel_theorem.
- Compact_space wikiPageWikiLink Heine–Borel_theorem.
- Compact_space wikiPageWikiLink Henri_Lebesgue.
- Compact_space wikiPageWikiLink Hilbert_cube.
- Compact_space wikiPageWikiLink Hilbert_space.
- Compact_space wikiPageWikiLink Homeomorphic.
- Compact_space wikiPageWikiLink Homeomorphism.
- Compact_space wikiPageWikiLink Hyperreal_field.
- Compact_space wikiPageWikiLink Hyperreal_number.
- Compact_space wikiPageWikiLink Infinite_sequence.
- Compact_space wikiPageWikiLink Infinitesimal.
- Compact_space wikiPageWikiLink Integral_equation.