Fort space

Remarks

Compactness: in an open cover, one member $U$ contains $p$, and the finitely many points outside $U$ need finitely many more members. Hausdorff: two points $x, y \ne p$ are separated by $\set{x}$ and $\set{y}$, and $x \ne p$ is separated from $p$ by $\set{x}$ and $X \setminus \set{x}$. Let $X$ be uncountable and let $U_1, U_2, \dots$ be open sets containing $p$. Their intersection has countable complement, so it contains a point $x \ne p$. Then $X \setminus \set{x}$ is an open set containing $p$ that contains no $U_n$, so $p$ has no countable neighbourhood base. The Fort space is the one-point compactification of the discrete space $X \setminus \set{p}$. If $X$ is countably infinite, it is homeomorphic to $\set{0} \cup \set{1/n : n \ge 1}$ with the usual topology, which is metrizable.