The Urysohn lemma states the following:

Let *X* be a normal space; let *A* and *B* be closed subsets of *X*. Let [*a*, *b*] be a closed interval of the real line. Then there exists a continuous map ƒ:*X* → [*a*, *b* ] such that ƒ(*x*) = *a* for every *x* in *A* and ƒ(*x*) = *b* for every *x* in *B*.

The Urysohn lemma is actually equivalent to the Tietze Extension Theorem, which states that if *X* is a normal space and *C* is a closed subset of *X*, then:

(a) Any continuous map of *C* into the closed interval [*a*, *b* ] of **R** may be extended to a continuous map of all of *X* into [*a*, *b* ]

(b) Any continuous map of *C* into **R** may be extended to a continuous map of all of *X* into **R**.