Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Question

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?

Answer 1

Hint: Let $R:=C(X, \mathbb R)$, since for each $f\in R$, $f=\sqrt[3]{f}\cdot\sqrt[3]{f}^2$, we have $M_x^2=M_x$. Suppose $M_x$ is principal, say $M_x=(g)$, then $g=g^2h$ for a certain $h\in R$, thus $M_x=(gh)$ and $(gh)^2=gh$, which forces that $gh$ takes value $0$ or $1$ everywhere, in particular, $f$ vanishes in a neighborhood of $x$. If $x$ is not isolated, that leads a contradiction.