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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.--> | |||
The theorem states the following: | |||
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> | |||
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. | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Katetov-Tong insertion theorem}} | |||
[[Category:General topology]] | |||
[[Category:Theorems in topology]] | |||
Revision as of 23:48, 6 November 2013
The Katětov–Tong insertion theorem is a theorem of point-set topology proved independently by Miroslav Katětov[1] and Hing Tong[2] in the 1950s.
The theorem states the following:
Let be a normal topological space and let be functions with g upper semicontinuous, h lower semicontinuous and . There exists a continuous function with
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.
References
- ↑ Miroslav Katětov, On real-valued functions in topological spaces, Fundamenta Mathematicae 38 (1951), 85–91. [1]
- ↑ Hing Tong, Some characterizations of normal and perfectly normal spaces, Duke Mathematical Journal 19 (1952), 289–292. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.