Proposition 4

Let $X$ be a compact space, let $A$ be a closed subspace of $X$, and let $H$ be a set of continuous real-valued functions on $X$ which separates the points of $C_A$ and is such that $A$ is the intersection of the sets $\overline{u}^{-1}(0)$ as $u$ runs through $H$. Then every continuous real-valued function on $X$ which is zero on $A$ can be uniformly approximated by polynomials without constant terms in the functions of $H$.

Consider first the particular case in which $A$ consists of a single point $x_0$. The hypotheses then imply that $H$ separates the points of $X$; for, if $x \neq x_0$, then by hypothesis there is a function $u \in H$ such that $u(x) \neq 0 = u(x_0)$. Hence, for each $\varepsilon > 0$ and each continuous real-valued function $f$ on $X$ such that $f(x_0) = 0$, there exists (Theorem 3) a polynomial $g$ in the functions of $H$ such that $|f(x) - g(x)| \leq \varepsilon$ for all $x \in X$. In particular $|g(x_0)| \leq \varepsilon$, so that

for all $x \in X$; and since $g(x) - g(x_0)$ is a polynomial in the functions of $H$ with no constant term, the result is established in this case.

In the general case, consider the equivalence relation $R$ on $X$ whose classes are the set $A$ and the sets $\{x\}$ for $x \notin A$. The quotient space $X/R$ is Hausdorff (Chapter I, § 8, no. 6, Proposition 15) and therefore compact. Let $\varphi : X \to X/R$ be the canonical mapping. Every continuous real-valued function $f$ on $X$ which vanishes on $A$ can be written in the form $f = f_1 \circ \varphi$, where $f_1$ is a continuous real-valued function on $X/R$ which vanishes at the point $x_0' = \varphi(A)$. Applying the result already proved to the space $X/R$ and the point $x_0'$, we obtain the final result.