Theorem 4

Let $(X_i)_{i \in I}$ be a family of compact spaces, and let

Then every continuous real-valued function on $X$ can be uniformly approximated by sums of a finite number of functions of the form where $J$ is an (arbitrary) finite subset of $I$ and $u_\alpha$ is a continuous real-valued function on $X_\alpha$ for each $\alpha \in J$.

Consider the set $H$ of "functions of one variable" $(x_i) \to u_\alpha(x_\alpha)$ (any $\alpha \in I$) which are continuous on $X$. This set separates the points of $X$, for if $x = (x_i)$ and $y = (y_i)$ are any two distinct points of $X$, there exists $\alpha \in I$ such that $x_\alpha \neq y_\alpha$ and there exists a continuous real-valued function $h_\alpha$ on $X_\alpha$ such that $h_\alpha(x_\alpha) \neq h_\alpha(y_\alpha)$. The function $x \to h_\alpha(\operatorname{pr}_\alpha x)$ then belongs to $H$ and takes distinct values at $x$ and $y$. Since every polynomial in the functions of $H$ is of the form stated in the theorem, the result follows from Theorem 3.

If not all the $X_i$ are compact, the conclusion of Theorem 4 is not necessarily valid (cf. Exercise 9).