If $X$ is compact Hausdorff and $p\in X$, then there is a continuous $f:X\to\mathbb{R}$ that vanishes at $p$ and nowhere else Prove or refute:  If $p$ is a point in a compact Hausdorff space $X$ then there exists a continuous real-valued function $f:X\to\mathbb{R}$ that vanishes at $p$ and nowhere else.

HINT: Consider the point $\omega_1$ in the space $\omega_1+1$ with the order topology. This question should help.
Added: Here’s a simpler example. Let $X$ be an uncountable set, and let $p\in X$. A set $U\subseteq X$ is open if and only if either $p\notin U$, or $X\setminus U$ is finite. Show that $X$ is a compact Hausdorff space, and that if $f:X\to\Bbb R$ is continuous, and $f(p)=0$, then there are uncountable many points $x\in X$ such that $f(x)=0$.
