A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus F\in\tau,$ there are disjoint sets $U,V\in\tau$ such that $E\subseteq U$ and $F\subseteq V.$
We say that $A,B\subseteq X$ are separated by a continuous function if there is a continuous function $f:X\to\Bbb R$ (with $\Bbb R$ considered in the usual topology) such that $A\subseteq f^{-1}\bigl[\{0\}\bigr]$ and $B\subseteq f^{-1}\bigl[\{1\}\bigr]$
Urysohn's Lemma says that a topological space is normal if and only if any two disjoint closed sets are separated by a continuous function. All well and good, but in a setting without (sufficiently strong) Choice principles, Urysohn's Lemma may fail to hold, as shown in this paper, by Good and Tree.
Has anyone encountered any other name for spaces in which disjoint closed sets are separated by continuous functions? Incidentally, I know that if sets are separated by continuous functions, then they are separated by open sets, so such spaces will be normal. I contemplated calling such spaces "completely normal," but this might lead to confusion, as hereditarily normal spaces (spaces such that every subspace is normal) are often referred to as completely normal. Any alternate suggestions?