Definition:Completely Regular Space

Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.

$\left({X, \vartheta}\right)$ is a completely regular space iff:


 * For any closed set $F \subseteq X$ and any point $y \in X$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\left\{{y}\right\}$.

Variants of Name
Earlier (pre-1970) treatment of this subject tends to refer to this as a $T_{3 \frac 1 2}$ space, and what we define as a $T_{3 \frac 1 2}$ (Tychonoff) space as a completely regular space.