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.