TITLE: What is the Krull dimension of $C(X)$ for $X$ infinite, compact and Hausdorff?
QUESTION [5 upvotes]: What is the Krull dimension of the ring of continuous real-valued functions on an infinite compact Hausdorff space?

If the Krull dimension is not finite, we will say that it is infinite, and not try to define it as a cardinal or an ordinal.
Of course the Krull dimension of $C(X)$ depends a priori on $X$, but I'd be very happy if it could be computed even in the most particular cases.

REPLY [3 votes]: The Krull dimension of $C(X)$  is infinite for any infinite compact Hausdorff space $X$.  By my answer here, it suffices to show there exists a continuous function $f:X\to\mathbb{R}$ which is not locally constant.  To show this, let $A=\{x_n\}$ be a countably infinite discrete subset of $X$.  For each $n$, by Urysohn's lemma there exists a continuous function $f_n:X\to [0,1/2^n]$ such that $f_n(x_m)=0$ for $m\geq n$ and $f_n(x_m)=1/2^n$ for $m<n$.  The sum $\sum f_n$ then converges uniformly on $X$ to a continuous function $f:X\to[0,2]$, and $f$ is injective when restricted to $A$.  Now let $y\in X$ be any accumulation point of $A$.  Since $f$ is injective on $A$ and every neighborhood of $y$ contains infinitely many points of $A$, $f$ is not constant in any neighborhood of $y$.