fully T4

A topological space $X$ is said to be fully $T_4$ if every open cover of $X$has star refinement.

A topological space is said to be fully normal if it is a $T_1$ space andis fully $T_4$.

For example, every pseudometric space is fully $T_4$.

We have the following implications:

Lindelöf $T_3\Rightarrow$ paracompact and $T_3\Rightarrow$ fully $T_4\Rightarrow T_4\Rightarrow$ uniformizable $\Rightarrow T_3$ ,

and

fully normal $\iff$ paracompact regular.