About: Heine–Cantor theorem

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In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if is a continuous function between two metric spaces and , and is compact, then is uniformly continuous. An important special case is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.

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rdfs:label	Teorema de Heine-Cantor (ca) Cantorova–Heineova věta (cs) Satz von Heine (de) Heine–Cantor theorem (en) Teorema de Heine-Cantor (es) Théorème de Heine (fr) Teorema di Heine-Cantor (it) 하이네-칸토어 정리 (ko) ハイネ・カントールの定理 (ja) Stelling van Heine-Cantor (nl) Twierdzenie Heinego-Cantora (pl) Теорема о равномерной непрерывности (ru) Heine–Cantors sats (sv) Теорема Кантора — Гейне (uk) 海涅－康托尔定理 (zh)
rdfs:comment	En matemàtiques, el teorema de Heine-Cantor, anomenat així per deure's a Eduard Heine i Georg Cantor, estableix que, si és una funció contínua entre dos espais mètrics i és compacte, llavors és uniformement contínua. (ca) V matematice Cantorova–Heineova věta, pojmenována po Georgu Cantorovi a Eduardovi Heineovi, říká, že pokud M je kompaktní metrický prostor, potom každá spojitá funkce f : M → N, kde N je metrický prostor, je stejnoměrně spojitá. Například, pokud f : [a,b] → R je spojitá funkce, pak je taktéž stejnoměrně spojitá. Toto není Cantorova věta. (cs) Der Satz von Heine (nach Eduard Heine; oder auch Satz von Heine-Cantor) aus der reellen Analysis macht eine Aussage über stetige Funktionen. Er wurde 1872 von Eduard Heine bewiesen und nach ihm benannt, nach Aussage von wurde diese Tatsache jedoch schon zuvor von Karl Weierstraß entdeckt. (de) En matemáticas, el teorema de Heine-Cantor, llamado así por deberse a Eduard Heine (1821 - 1881) y Georg Cantor, establece que, si f : M → N es una función continua entre dos espacios métricos y M es compacto, entonces f es uniformemente continua. (es) In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if is a continuous function between two metric spaces and , and is compact, then is uniformly continuous. An important special case is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous. (en) Le théorème de Heine, démontré par Eduard Heine en 1872, s'énonce ainsi : toute application continue d'un espace métrique compact dans un espace métrique quelconque est uniformément continue. Cela implique notamment que toute fonction continue d'un segment [a, b] dans ℝ est uniformément continue. (fr) ハイネ・カントールの定理（英語: Heine–Cantor theorem）とは、次のような定理である。 M をコンパクトな距離空間、N を距離空間とする。このとき、任意の連続関数 f : M → N は一様連続である。 (ja) In matematica, il teorema di Heine - Cantor è un teorema di analisi matematica riguardante l'uniforme continuità di funzioni definite fra spazi metrici. Prende il nome da Eduard Heine e Georg Cantor. In generale ogni funzione uniformemente continua è anche continua. Il teorema di Heine-Cantor permette di invertire tale implicazione, nell'ipotesi che il dominio sia uno spazio metrico compatto. (it) 해석학에서 하이네-칸토어 정리(Heine-Cantor定理, 영어: Heine–Cantor theorem)는 두 균등 공간 사이의 함수에 대하여, 만약 정의역이 콤팩트 공간이라면 연속 함수의 개념과 균등 연속 함수의 개념이 일치한다는 정리다. (ko) In de wiskunde stelt de stelling van Heine-Cantor, genoemd naar Eduard Heine en Georg Cantor, dat elke continue functie van een compacte metrische ruimte naar een metrische ruimte ook uniform continu is. Als bijvoorbeeld een continue functie is, dan is ook uniform continue. (nl) Twierdzenie Heinego-Cantora – nazwane na cześć Heinricha Heinego oraz Georga Cantora twierdzenie mówiące że każda funkcja ciągła na przestrzeni zwartej jest jednostajnie ciągła. (pl) Heine–Cantors sats är en matematisk sats uppkallad efter Georg Cantor och Eduard Heine som säger att om M är ett kompakt metriskt rum är varje kontinuerlig funktion , där N är ett metriskt rum, likformigt kontinuerlig. (sv) Теорема о равномерной непрерывности или Теоре́ма Ка́нтора — Ге́йне говорит, что непрерывная функция, определённая на компакте, равномерно непрерывна на нём. (ru) Теорема Кантора — Гейне в математичному і функціональному аналізі стверджує, що функція, неперервна на компакті, рівномірно неперервна на ньому. (uk) 海涅-康托尔定理，以愛德華·海涅和乔治·康托尔命名，说明如果M是一个紧度量空间，N是一个度量空间，则每一个连续函数 f : M → N, 都是一致连续的。特别地，如果f : [a,b] → R是一个连续函数，则它是一致连续的。 (zh)
differentFrom	Cantor's theorem
dcterms:subject	Articles containing proofs Theorems in analysis Continuous mappings Metric geometry
Wikipage page ID	1194729 (xsd:integer)
Wikipage revision ID	1084933155 (xsd:integer)
Link from a Wikipage to another Wikipage	Articles containing proofs Compact space Continuous function Mathematics Georg Cantor Bounded set Triangle inequality Domain of a function Eduard Heine Non-standard calculus Closed interval Interval (mathematics) Cover (topology) Theorems in analysis Continuous mappings Metric geometry Metric space Open set Real number Set (mathematics) Uniform continuity Uniformly continuous
sameAs	Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem Heine–Cantor theorem
dbp:wikiPageUsesTemplate	dbt:Distinguish dbt:No_footnotes dbt:Planetmath
title	Heine–Cantor theorem (en)

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