What class of topological spaces is this? It is well-known but apparently I have forgotten the answer to the below question. Which topological spaces $V$ have this property:

For every open subset $U\subset V$ there exists a non-zero continuous function $f\colon V\to \mathbb{R}$ such that $\overline{\{x\in V\colon f(x)\neq 0\}}\subset U$?

Is it normality by chance?
 A: Every completely regular space has your property, whether or not it is $T_1$. For each $x\in X$ and open nbhd $U$ of $x$, complete regularity gives you a continuous $f_{x,U}:X\to[0,1]$ such that $f_{x,U}(x)=1$ and $f_{x,U}(y)=0$ for all $y\in X\setminus U$. Now let $U$ be any non-empty open set in the completely regular space $X$. Fix $x\in U$ arbitrarily, and let $$V=\left\{y\in U:f_{x,U}(y)>\frac12\right\}\;;$$ then $V$ is an open nbhd of $x$, and $f_{x,V}$ is a continuous real-valued function on $X$ whose support is contained in $U$.
However, there are also spaces with your property that are not completely regular. Let $X$ be the space constructed by John Thomas in ‘A Regular Space, Not Completely Regular’, The American Mathematical Monthly, Vol. $76$, No. $2$ (Feb., $1969$), pp. $181$-$182$, and described in detail in this answer. $X$ has a dense set of isolated points, so if $U$ is a non-empty open set in $X$, $x\in U$ be an isolated point, and define
$$f:X\to\Bbb R:y\mapsto\begin{cases}
1,&\text{if }y=x\\
0,&\text{otherwise}\;;
\end{cases}$$
clearly $f$ is continuous and has support $\{x\}\subseteq U$.
A: This is implied by complete regularity + $T_1$, also called Tychonoff, a property weaker than normality.
If $U$ is a (non-empty) open set, find a non-empty open $V$ such that $\overline{V} \subset U$ (by regularity) and for some $p \in V$ a continuous function $f: V \rightarrow [0,1]$ with $f(p) = 1$ and $f([X \setminus V] = \{0\}$, by complete regularity. Then $\overline{\left\{x \in V: f(x) \neq 0\right\}} \subset \overline{V} \subset U$, as required.
It doesn't seem to be equivalent to it, as Brian pointed out. But it is true in all Tychonoff ($T_{3\frac{1}{2}}$) spaces.
