Katětov–Tong insertion theorem

The Katětov–Tong insertion theorem is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s.

The theorem states the following:

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

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