5
$\begingroup$

This question was created in a discussion.

Let $X$ be a topological space. Denote by $C(X; \mathbb{R})$ the ring of real-valued continuous functions defined on $X.$

Characterize those compact Hausdorff spaces $X$ for which $C(X; \mathbb{R})$ has the property that all of its prime ideals are maximal.

$\endgroup$
10
$\begingroup$

These are exactly the finite spaces.

Completely regular topological spaces $X$ with the property that every prime ideal in $C(X)$ is maximal are called $P$-spaces. A topological characterization is: every $G_{\delta}$-set (countable intersection of open sets) is itself open.

It is not hard to show that countable subsets of $P$-spaces are closed and discrete. It follows that countably compact and in particular compact $P$-spaces are finite.

For a long list of equivalent characterizations and basic properties of $P$-spaces consult exercises 4J and 4K on pages 62 and 63 of Gillman-Jerison, Rings of continuous functions, Springer GTM 43, 1976.

In the MO-thread on prime ideals in $C[0,1]$ you'll also find some relevant information and constructions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.