Definition 5.24.2 (Separation of Points and Closed Sets).label Let $X$ be a topological space and $\cf \subset C(X; [0, 1])$, then $\cf$ separates points and closed sets if for any $E \subset X$ closed and $x \in X \setminus E$, there exists $f \in \cf$ such that $f(x) \not\in \ol{f(E)}$.