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 and is 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 $\Leftrightarrow$ paracompact regular.