(Stone-Weierstrass) Let $K$ be compact HAusdorff and $A\subset C_\mathbb{R}(K)$ a (sub)algebra which sepearates points. Then either $\overline{A}=C_\mathbb{R}(K)$ or $\exists x_0$ such that $\overline{A}=\{ f\in C_\mathbb{R}(K): f(x_0)=0\}$
I have seen lots of applications of the Stone-Weierstrass theorem to prove that a set is dense in $C(\mathbb{R})$. This is one possible conclusion of the theorem. The other is that if the algebra $A$ satisfies the hypothesis, then $\overline{A}=\{\text{all continuous functions which vanish at some } x_0\}$
What applications use this second result of the theorem, or is it just the degenerate case?