# Urysohn lemma

For any two disjoint closed sets $A$ and $B$ of a normal space $X$ there exists a real-valued function $f$, continuous at all points, taking the value $0$ at all points of $A$, the value 1 at all points of $B$ and for all $x\in X$ satisfying the inequality $0\leq f(x)\leq1$. This lemma expresses a condition which is not only necessary but also sufficient for a $T_1$-space $X$ to be normal (cf. also Separation axiom; Urysohn–Brouwer lemma).