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This example is the book Functional Analysis by Walter Rudin in page 288 Exercise 3.

If $X$ is a compact Hausdorff space, show that there is a natural one-to-one correspondence between closed subset $X$ and closed ideals of $C(X)$.

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    @Matema: Since you are still new here, I want to give you some advice about the site: **To get the best possible answers, you should explain what your thoughts on the problem are so far**. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. Also, many would consider your post rude because it is a command ("Show..."), not a request for help, so please consider rewriting it.2012-12-28

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I guess $C(X)$ is endowed with the natural norm, that is $\lVert f\rVert:=\sup_{x\in X}|f(x)|$. Let $\mathcal F$ the collection of closed subsets of $X$ and $\mathcal I$ the collection of the closed ideals of $C(X)$. We can define $\phi(F):=\{f\in C(X),\forall x\in F,\, f(x)=0\}.$ What we have to show it that

  • $\phi(F)$ is an ideal of $C(X)$;
  • $\phi(F)$ is closed;
  • given two distinct closed sets $F_1$ and $F_2$, if $F_1$ is not contained in $F_2$ we can find a continuous function which vanishes on $F_1$ but not on $F_2$;
  • given a closed ideal $I$, we can find a closed set $F$ such that $\phi(F)=I$.