Katetov-Tong insertion theorem

From Wikipedia, the free encyclopedia

You have new messages (last change).

The Katetov-Tong insertion theorem is a theorem of point-set topology originally proved by Hing Tong in the 1950s. Miroslav Katetov shares the name for a revised version of the theorem and an improved proof.

The theorem states the following:

Let X be a normal topological space and let g, h : X \to \mathbb{R} be functions with g upper semicontinuous, h lower semicontinuous and g \leq h. There exists a continuous function f : X \to \mathbb{R} with g \leq f \leq h.

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.

Retrieved from "http://en.wikipedia.org../../../k/a/t/Katetov-Tong_insertion_theorem_0fed.html"