Can the Stone–Weierstrass theorem be applied in two variables metric space X × Y?

Question

If $X$ and $Y$ are two compact spaces and view $X\times Y$ as a metric space with metric $d((x,y),(x',y'))=\sqrt{dx(x,x')^2+dy(y,y')^2}$, $f \colon X \times Y \to \Bbb R$ is a continuous function, then for every $\epsilon > 0$ there exist $n > 0$ and continuous functions  $f_1,\dots,  f_n$  on $X$ and continuous functions $g_1, \dots, g_n$ on $Y$ such that $\|f − ∑ f_i g_i\| < \epsilon$.

Do I first need to prove that $X\times Y$ is also a compact metric space? how can I do this from the given metric?

And is stone-weierstrass only suitable for one variable, how can I apply it for two variables $x$ and $y$?

Answer 1

1. Show that $d$ is a metric on $X\times Y$ (this is straight forward.)
2. The product of compact spaces is compact (well known topological fact.)
3. Let $A$ be the set of all functions on $X\times Y$ of the form $\sum_{i=1}^nf_i(x)\,g_i(y)$ where the $f_i$ are continuous on $X$ and the $g_i$ continuous on $Y$.

4. Show that $A$ satisfies the conditions of the Stone-Weirstrass theorem:

   • $A$ is an algebra.

   • $A$ contains the constants.

   • $A$ separates points.