Let $X$ be a compact Hausdorff space and $C(X)$ be the algebra of all continuous functions on $X$ equipped with sup-norm. I am looking for an ideal in $C(X)$ which is not closed. I know that there is a bijection between the collection of closed sets in $X$ and the collection of closed Ideals in $C(X)$ .
2026-03-25 07:39:49.1774424389
Characterizing Ideals in C(X)
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Well there may not be any; for example if $X$ is finite then every ideal in $C(X)$ is closed. (I suspect that $X$ finite is the only time that happens, haven't really thought about it.)
If for example $X=[0,1]$ you could let $I$ be the set of all $f$ of the form $f(t)=tg(t)$ for some continuous function $g$. If that were a clsoed ideal it would have to be the set of all $f$ with $f(0)=0$, but it's not, for example consider $f(t)=\sqrt t$.