Urysohn's lemma

This article gives the statement, and possibly proof, of a basic fact in topology.

Statement

Let $X$ be a Normal space (?) (i.e., a topological space that is T1 and where disjoint closed subsets can be separated by disjoint open subsets). Suppose $A,B$ are disjoint closed subsets of $X$. Then, there exists a continuous function $f:X\to [0,1]$ such that $f(a)=0$ for all $a\in A$, and $f(b)=1$ for all $b\in B$.

Note that the T1 assumption is not necessary, so Urysohn's lemma also holds for normal-minus-Hausdorff spaces, which is what many point set topologists are referring to when they use the term normal space.

Related facts

• Tietze extension theorem
• Hahn-Dieudonne-Tong insertion theorem