Existence of a special type of maximal ideal in $C(X)$:

Question

Does there exist any maximal ideal $M^p$ in $C(X)$ (the ring of continuous functions on a topological space $X$) such that each element of $M^p$ is a divisor of zero but $M^p≠O^p$?

Answer 1

Assuming that $M^p$ is the set of elements of $C(X)$ which vanish at $p$, and $O^p$ is the set of elements of $C(X)$ which vanish on a neighborhood of $p$, the answer depends on the space $X$. For example, if $X = [0,1]$, since every singleton is a zero-set, there is no such maximal ideal. On the other hand if $X = D \cup \{\infty\}$ is the one-point compactification of the discrete space of size $\aleph_1$ and $p = \infty$, then $M^p \ne O^p$ but each element of $M^p$ is a zero divisor. The reason is that if $\{x_n : n = 1, 2, \cdots\}$ is a countable subset of $D$, the function $f \in C(X)$ given by $f(x_n) = \frac{1}{n}$ and $f(x) = 0$ otherwise is not $0$ on any neighborhood of $p$ (because neighborhoods of $p$ are cofinite). On the other hand, since zero-sets are G$_\delta$ sets and $\{\infty \}$ is not a G$_\delta$ set, every element of $C(X)$ which vanishes at $\infty$ also vanishes at an isolated point and, therefore, is a zero-divisor.