I have just learned Stone's theorem (at least as is written in baby Rudin):
Let $\mathscr{A}$ be an algebra of real continuous functions on a compact set $K$. If $\mathscr{A}$ separates points on $K$ and if $\mathscr{A}$ vanishes at no point of $K$, then the uniform closure $\mathscr{B}$ of $\mathscr{A}$ consists of all real continuous functions on $K$.
I was wondering about concrete examples of algebras $\mathscr{A}$ of real continuous functions on compact sets that separate points and do not vanish. Obviously one choice is the set of all polynomials on $K$, and this case yields Weierstrass' theorem. What are some other examples? Are there nice examples when $K$ is a compact subset of $\mathbb{R}$?