Lemma 11

Let X be a completely regular topological space, that is to say uniformizable and separated (TG, IX, p. 8, Definition 4), containing at least two points. There exist non-zero continuous functions $f$ and $g$ in $\mathscr{C}(X)$ such that $f g= 0$.

Let $x\not=y$ be distinct points of X. Let U and V be open neighbourhoods of $x$ and $y$, respectively, such that U and V are disjoint. Since X is uniformizable, by TG, IX, p. 7, Theorem 2, there exists a function $f\in \mathscr{C}(X)$ such that $f(x) = 1$ and $f|X$ - U = 0. Analogously, there exists $g\in \mathscr{C}(X)$ such that $g(y) = 1$ and $g|X$ - V = 0. One then has $f g= 0$.