# Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

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?

• I think it would be systematic to use “completely normal” in the same meaning as “completely regular” and “completely Hausdorff” and use “hereditarily normal” for the other notion. In the context where AC might not hold, many things are different and a reader might be careful anyway, so why not to declare these namings in this context (e.g. in a paper)? – user87690 Aug 14 '13 at 7:40
• @user87690 "completely normal" is already used for hereditarily normal (it requires a small proof that they are equivalent notions: completely normal is defined as the fact that we can separate any two separated sets of a space using open sets. Sets are separated if the closure of each misses the other. – Henno Brandsma Jun 15 '14 at 6:55
• Maybe "functionally normal" would do. Functionally Hausdorff is sometimes used for being able to separate points by real-valued continuous functions. – Henno Brandsma Jun 15 '14 at 7:26
• @HennoBrandsma But then it would be better to use “functionally regular” instead of “completely regular” and “functionally Hausdorff” instead of “completely Hausdorff”. And “completely Hausdorff” should mean that separated points can be separated by open sets and the same for “completely regular”. This seems to be more invasive change than simply stop to use “completely normal” in sense of “hereditarily normal”. – user87690 Jun 15 '14 at 7:45
• @Henno: I know that a space is hereditarily normal iff separated sets can be separated by open sets. The problem is that the latter needn't fit with other "completely [descriptor]" forms if we don't have Choice, and I want to keep it consistent. I like "functionally [descriptor]" as a substitute. Feel free to make that an answer, and (if possible) indicate some source(s) in which you've seen that. – Cameron Buie Jun 15 '14 at 16:33