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 $C(X)$.
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 $C(X)$.
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