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Compact space

In mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value infinitely often, by the pigeonhole principle. For subsets of Euclidean space, the analogous statement is sequential…

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Overview

Historical development

Basic examples

Definitions

Sufficient conditions

Properties of compact spaces

Examples

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Map overview Semantic statistics

Compact space

Nodes188
Edges187
Triples106
Avg. degree1.99
Density0.010638
Components1

How this topic connects Entity context

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Compact space

Top relations

related to Examples · 39
Compact space → Again, Alaoglu, Alaoglu's, Alexandroff, Any, Arzelà, Ascoli, Banach, Borel, Cantor, Consider, Conversely, Euclidean, Finite, For, Hausdorff, Heine, Heine-Borel, Here, Hilbert
related to Historical development · 22
Compact space → Arzelà, Ascoli, Bernard Bolzano, Bolzano, Bolzano's, Cesare Arzelà, David Hilbert, Erhard Schmidt, For, Giulio Ascoli, Green's, Hilbert, In, It, Karl Weierstrass, Maurice Fréchet, On, Schmidt, The, This
related to Basic examples · 9
Compact space → Any, For, However, If, In, It, Likewise, Lines, The
related to External links · 7
Compact space → Creative Commons Attribution/Share-Alike License, Examples, HO, Manya Raman, PlanetMath, Sundström, This
related to Properties of compact spaces · 7
Compact space → Hausdorff, Hausdorff TVS, However, If, In, KC, TVS
related to Sufficient conditions · 6
Compact space → Hausdorff, If, In, The, This, Tychonoff's
related to Compactifications · 4
Compact space → Alexandroff, By, Every, Hausdorff
related to Functions and compact spaces · 3
Compact space → As, Since, Slightly
see also · 2
Compact space → Compactly, Hausdorff

Important terminology Word statistics

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Important terminology

compact space every closed subset finite set compactness theorem sequence open hausdorff bounded topology point spaces subsets topological cover interval

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
the open intervalinstance ofwhereas this fails for spaces0.80text
the Arzelàinstance ofand major results0.80text
algebraic geometryinstance ofSome branches of mathematics0.80text
typically influenced by the French school of Bourbakiinstance ofSome branches of mathematics0.80text
use the term quasi-compact for the general notioninstance ofSome branches of mathematics0.80text
and reserve the term compact for topological spaces that are both Hausdorffinstance ofSome branches of mathematics0.80text
quasi-compactinstance ofSome branches of mathematics0.80text
Compact spacerelated to Basic examplesAny0.60section
Compact spacerelated to Basic examplesIf0.60section
Compact spacerelated to Basic examplesFor0.60section
Compact spacerelated to Basic examplesThe0.60section
Compact spacerelated to Basic examplesIt0.60section

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    Min side: 3
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