Urysohn's lemma

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalized by (and usually used in the proof of) the Tietze extension theorem.

The lemma is named after the mathematician Pavel Samuilovich Urysohn.

Formal statement

Two disjoint closed subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhoods U of A and V of B that are also disjoint. A and B are said to be separated by a function if there exists a continuous function f from X into the unit interval [0,1] such that f(a) = 0 for all a in A and f(b) = 1 for all b in B. Any such function is called a Urysohn function for A and B.

A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.