Show that there is a natural one-to-one correspondence

Question

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 $C(X)$.

Answer 1

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 $\phi(F)=I$.