A family of continuous unit-interval-valued functions that separates points from closed sets

Let $X$ be a topological space. A family $\mathcal F$ of continuous maps $f:X\to[0,1]$ (Continuity of a map of topological spaces at a point and globally, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) separates points from closed sets when both conditions hold:

1. for every distinct $x,y\in X$, some $f\in\mathcal F$ has $f(x)\ne f(y)$; and
2. for every closed $C\subseteq X$ and every $x\notin C$, some $f\in\mathcal F$ satisfies $f(x)=1$ and $f[C]=\{0\}$.

The empty family has these properties precisely when $X=\varnothing$. Indeed, if $x\in X$, the closed set $C=\varnothing$ makes clause 2 demand a member of the family. Thus a one-point space still needs a separating function for its point and the empty closed set; no coordinate is silently selected in either case.