Definition:Tychonoff Space

Definition
Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a Tychonoff Space or completely regular space :
 * $\struct {S, \tau}$ is a $T_{3 \frac 1 2}$ space
 * $\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space.

That is:


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


 * $\forall x, y \in S$, either:
 * $\exists U \in \tau: x \in U, y \notin U$
 * $\exists U \in \tau: y \in U, x \notin U$

Also known as
Some sources give this as $T_{\frac 7 2}$ space, which of course evaluates to the same as a $T_{3 \frac 1 2}$ space.