Definition:Tychonoff Space

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

$\left({X, \vartheta}\right)$ is a Tychonoff Space or $T_{3 \frac 1 2}$ space iff:
 * $\left({X, \vartheta}\right)$ is a completely regular space
 * $\left({X, \vartheta}\right)$ is a Kolmogorov ($T_0$) space.

That is:


 * 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 $\left\{{y}\right\}$.


 * $\forall x, y \in X$, either:
 * $\exists U \in \vartheta: x \in U, y \notin U$
 * $\exists U \in \vartheta: y \in U, x \notin U$

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