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| The '''Katětov–Tong insertion theorem''' is a [[theorem]] of [[point-set topology]] proved independently by [[Miroslav Katětov]]<ref>Miroslav Katětov, ''On real-valued functions in topological spaces'', Fundamenta Mathematicae 38 (1951), 85–91. [http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=38]</ref> and [[Hing Tong]]<ref>Hing Tong, ''Some characterizations of normal and perfectly normal spaces'', Duke Mathematical Journal 19 (1952), 289–292. {{doi|10.1215/S0012-7094-52-01928-5}}</ref> in the 1950s. <!--[[Miroslav Katětov]] shares the name for a revised version of the theorem and an improved proof.-->
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| The theorem states the following:
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| Let <math>X</math> be a normal [[topological space]] and let <math>g, h\colon X \to \mathbb{R}</math> be functions with g upper [[semicontinuous]], h lower semicontinuous and <math>g \leq h</math>. There exists a continuous function <math>f\colon X \to \mathbb{R}</math> with <math>g \leq f \leq h.</math>
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| This theorem has a number of applications and is the first of many classical insertion theorems. In particular it implies the [[Tietze extension theorem]] and consequently [[Urysohn's lemma]], and so the conclusion of the theorem is equivalent to normality.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Katetov-Tong insertion theorem}}
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| [[Category:General topology]]
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| [[Category:Theorems in topology]]
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Revision as of 20:18, 16 February 2014
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