Urysohn's Lemma - Formal Statement

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 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.

The sets A and B need not be precisely separated by f, i.e., we do not, and in general cannot, require that f(x) ≠ 0 and ≠ 1 for x outside of A and B. This is possible only in perfectly normal spaces.

Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal T₁ spaces are Tychonoff.