Definition:Fully Normal Space

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

$T$ is fully normal iff:
 * $T$ is fully $T_4$
 * $T$ is a $T_1$ (Fréchet) space.

That is, $T$ is fully normal iff:
 * Every open cover of $X$ has a star refinement
 * All points of $T$ are closed.

Also see

 * A fully normal space is also a normal space.