Urysohn's lemma is the moment normality turns into continuous real-valued functions. It is the technical heart behind metrization and extension theorems.
Normal separation by functions
Normality separates closed sets by open neighborhoods; Urysohn upgrades that separation to a continuous function.
Theorem. If \(X\) is normal and \(A,B\subseteq X\) are disjoint closed sets, then there exists a continuous function \(f:X\to[0,1]\) such that \(f(A)=0\) and \(f(B)=1\).
Proof. The construction in the next two sections proves the theorem. Normality first gives nested open sets \(U_r\) indexed by dyadic rationals with \(\overline{U_r}\subseteq U_s\) for \(r<s\). Defining \(f(x)=\inf\{r:x\in U_r\}\) then yields \(f(A)=0\), \(f(B)=1\), and continuity from the openness of the level sets \(\{f<a\}\) and \(\{f>a\}\).
Remark. The conclusion is false in general Hausdorff spaces. Normality is exactly the open-set supply needed to build a continuous scale between two closed sets.
Nested open sets
The proof builds open sets indexed by dyadic rationals, with closures nested in the correct order.
Lemma. If \(X\) is normal, \(F\subseteq U\), \(F\) is closed, and \(U\) is open, then there is an open set \(V\) with \(F\subseteq V\subseteq\overline V\subseteq U\).
Proof. The closed sets \(F\) and \(X\setminus U\) are disjoint. Normality gives disjoint open sets \(V\supseteq F\) and \(W\supseteq X\setminus U\). Since \(W\) is disjoint from \(V\), the closure \(\overline V\) is disjoint from \(X\setminus U\), hence contained in \(U\).
Proposition. There are open sets \(U_r\), indexed by dyadic rationals \(r\in[0,1]\), such that \(A\subseteq U_0\), \(X\setminus U_1\supseteq B\), and \(r<s\) implies \(\overline{U_r}\subseteq U_s\).
Proof. Start with \(U_1=X\setminus B\). Use the lemma to choose \(U_0\) with \(A\subseteq U_0\subseteq\overline{U_0}\subseteq U_1\). Insert dyadic levels inductively: whenever \(r<t\) are adjacent at a finite dyadic stage, choose \(U_s\) for the midpoint \(s\) so that \(\overline{U_r}\subseteq U_s\subseteq\overline{U_s}\subseteq U_t\).
The Urysohn function
The function is the first dyadic level whose open set contains the point.
Definition. Define \(f(x)=\inf\{r\in\mathbb D\cap[0,1]:x\in U_r\}\), where \(\mathbb D\) denotes the dyadic rationals.
Proposition. The function \(f\) is continuous, equals \(0\) on \(A\), and equals \(1\) on \(B\).
Proof. The endpoint values follow from \(A\subseteq U_0\) and \(B\cap U_r=\varnothing\) for all \(r<1\). For continuity, it suffices to show \(\{f<a\}\) and \(\{f>a\}\) are open. The first is the union of \(U_r\) over dyadic \(r<a\). The second is the union of \(X\setminus\overline{U_r}\) over dyadic \(r>a\), using the nesting relation to compare levels. Both are open.
Consequences
Urysohn functions separate topology by scalar-valued observables.
Corollary. Every normal \(T_1\) space is completely regular: points and closed sets can be separated by continuous functions into \([0,1]\).
Proof. Apply Urysohn's lemma to the disjoint closed sets \(\{x\}\) and \(F\).
Remark. This function-separation principle is the entrance to Tychonoff embeddings and Stone-?ech compactification.