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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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    What about the set of functions which vanish on this closed subset?2012-12-28
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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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