Definition 5.24.1 (Completely Regular).label Let $X$ be a topological space, then $X$ is completely regular if for any $E \subset X$ closed and $x \in X \setminus E$, there exists $f \in C(X; [0, 1])$ such that $f(x) = 1$ and $f|_{E} = 0$.
Definition 5.24.1 (Completely Regular).label Let $X$ be a topological space, then $X$ is completely regular if for any $E \subset X$ closed and $x \in X \setminus E$, there exists $f \in C(X; [0, 1])$ such that $f(x) = 1$ and $f|_{E} = 0$.
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