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.

I asked a very closely related question on this site a while back.  The accepted answer shows that a compact Hausdorff space has the property that every singleton set is the zero set of a continuous real-valued function iff it is first countable.  This gives a very clean answer to your question.
