Maximal ideals of C(X) requires compactness of X

Question

I have come across the standard Nulstellensatz kind of characterization of Maximal ideals of C(X). But that holds when X is a compact, Hausdorff space. Can you give me an example where the characterization does not hold for non compact spaces?

Answer 1

Yes take $(0,1)$ consider the maximal ideal containing the ideal of all functions that vanish in lets say in $(a,1)$ for some $a$.The maximal ideal containing this ideal is not of this form as you can always dodge a point of common $0$.

Answer 2

Take $X = \mathbb{N}$ and consider the continuous functions $f_F:X \rightarrow \mathbb{R}$ defined by $f_F(x) = 1$ if $x \in F$, $f_F(x) = 0$ for $x \notin F$, where $F$ ranges over all finite subsets of $\mathbb{N}$.

Let $I$ be any maximal ideal that contains all $\{f_F: F \subset \mathbb{N}\text{ finite }\}$. This can be done as they are contained in the ideal of real functions that vanish outside some finite set, and any ideal is contained in a maximal ideal.

Then for any $p \in \mathbb{N}$, $f_{\{p\}}(p) = 1 \neq 0$ and $f_{\{p\}} \in I$, so $I$ is not of the fixed-point form.

All such maximal ideals actually correspond to points of $\beta\mathbb{N}$ or equivalently free ultrafilters on $\mathbb{N}$, and it is well known (see Gillman and Jerrison's book Rings of Continuous functions) that "every maximal ideal $I$ on $C(X)$ is of the form $I_p = \{f \in C(X): f(p) = 0\}$ for some $p \in X$" actually characterises $X$ being compact within the class of Tychonoff spaces (ensuring that $C(X)$ has enough elements to be "meaningful").