Corollary 1

Let $X$ be a regular souslinian space, $H$ a set of continuous numerical functions in $X$, which separates the points of $X$ (IX, p. 9, def. 5). There then exists a countable subset $H'$ of $H$ which separates the points of $X$.

Indeed, $X \times X$ is a souslinian space (IX, p. 60, prop. 7); the same is true of every open subspace of $X \times X$ (IX, p. 59, prop. 5), which is therefore a Lindelöf space (IX, p. 76, corollary). To every function $h \in H$, associate the closed set $F_h$ of the pairs $(x, y) \in X \times X$ such that $h(x) = h(y)$. By virtue of prop. 3, there exists a countable subset $H'$ of $H$ such that one has $\bigcap_{h \in H'} F_h = \bigcap_{h \in H} F_h$. But by assumption the second member is the diagonal of $X \times X$; the same is therefore true of the first, and consequently $H'$ separates the points of $X$.