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I want a reference (or proof) for the following Stone-Weierstrass type Theorem stated in Wikipedia, Section 2.2 .

Theorem Let $X,\tilde X $ be compact Hausdorff topological spaces and $f\in C(X\times\tilde X)$, space of real-valued continuous functions on $X\times \tilde X$ equipped with the supremum norm. Then for all $\epsilon>0$ $\exists N,\ \{g_n\}_{n=1}^N\subset C(X),\ \{\tilde g_n\}_{n=1}^N\subset C(\tilde X)$ such that $$ \| f-\sum_{n=1}^N g_n\tilde g_n\|\le\epsilon. $$

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    see here http://www.ams.org/journals/proc/1981-081-01/S0002-9939-1981-0589143-8/S0002-9939-1981-0589143-8.pdf i hope this will help you – 2017-01-23

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What the result states is that the algebra $A$ of sums of products of elements in $C(X)$ and $C(X')$ is dense in $C(X\times X')$. Using the Stone-Weierstrass theorem, all you need to show is that $1\in A$ (which is trivial) and that $A$ separates points of $X\times X'$.

Suppose $(x_1,x_1'),(x_2,x_2')\in X\times X'$ are not equal. Then WLOG we can assume $x_1\neq x_2$. There is some $g\in C(X)$ such that $g(x_1)\neq g(x_2)$. Now define $f\in C(X\times X')$ by $f(x,x')=g(x)$. Then $f\in A$ ($f$ is the product of $g$ and the identity of $C(X')$), and $f(x_1,x_1')\neq f(x_2,x_2')$. Hence $A$ separates points of $X\times X'$, so the Stone-Weierstrass theorem applies and therefore $A$ is dense in $C(X\times X')$.

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    @Ton I'm not familiar with a proof, but I would recommend looking at the one-point compactification of the space. Then try to apply the regular Stone-Weierstrass theorem. – 2017-02-12
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    Thank you. The above comment of mine was just me mis-submitting a comment. I think I found the version for locally compact Hausdorff domains on a book from Willard ("general topology"). – 2017-02-12