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- Heine–Borel_theorem abstract "In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:S is closed and boundedevery open cover of S has a finite subcover, that is, S is compact.In the context of real analysis, the former property is sometimes used as the defining property of compactness. However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness. In fact, the Heine–Borel theorem for arbitrary metric spaces reads:A subset of a metric space is compact if and only if it is complete and totally bounded.".
- Heine–Borel_theorem wikiPageID "59595".
- Heine–Borel_theorem wikiPageRevisionID "573442808".
- Heine–Borel_theorem id "3328".
- Heine–Borel_theorem id "p/b017100".
- Heine–Borel_theorem title "Borel-Lebesgue covering theorem".
- Heine–Borel_theorem title "proof of Heine-Borel theorem".
- Heine–Borel_theorem subject Category:Articles_containing_proofs.
- Heine–Borel_theorem subject Category:Compactness_theorems.
- Heine–Borel_theorem subject Category:General_topology.
- Heine–Borel_theorem subject Category:Properties_of_topological_spaces.
- Heine–Borel_theorem subject Category:Theorems_in_real_analysis.
- Heine–Borel_theorem comment "In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:S is closed and boundedevery open cover of S has a finite subcover, that is, S is compact.In the context of real analysis, the former property is sometimes used as the defining property of compactness.".
- Heine–Borel_theorem label "Heine–Borel theorem".
- Heine–Borel_theorem label "Satz von Heine-Borel".
- Heine–Borel_theorem label "Stelling van Heine-Borel".
- Heine–Borel_theorem label "Teorema de Heine-Borel".
- Heine–Borel_theorem label "Teorema de Heine-Borel".
- Heine–Borel_theorem label "Teorema di Heine-Borel".
- Heine–Borel_theorem label "Théorème de Borel-Lebesgue".
- Heine–Borel_theorem label "Twierdzenie Heinego-Borela".
- Heine–Borel_theorem label "Лемма Гейне — Бореля".
- Heine–Borel_theorem label "مبرهنة هاين-بوريل".
- Heine–Borel_theorem label "ハイネ・ボレルの被覆定理".
- Heine–Borel_theorem label "海涅-博雷尔定理".
- Heine–Borel_theorem sameAs Heine%E2%80%93Borel_theorem.
- Heine–Borel_theorem sameAs Satz_von_Heine-Borel.
- Heine–Borel_theorem sameAs Teorema_de_Heine-Borel.
- Heine–Borel_theorem sameAs Théorème_de_Borel-Lebesgue.
- Heine–Borel_theorem sameAs Teorema_di_Heine-Borel.
- Heine–Borel_theorem sameAs ハイネ・ボレルの被覆定理.
- Heine–Borel_theorem sameAs 하이네-보렐_정리.
- Heine–Borel_theorem sameAs Stelling_van_Heine-Borel.
- Heine–Borel_theorem sameAs Twierdzenie_Heinego-Borela.
- Heine–Borel_theorem sameAs Teorema_de_Heine-Borel.
- Heine–Borel_theorem sameAs Q253214.
- Heine–Borel_theorem sameAs Q253214.
- Heine–Borel_theorem wasDerivedFrom Heine–Borel_theorem?oldid=573442808.