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.