Existence of a metric space where each open ball is closed and has a limit point Show that there exists a metric space in which every open ball is closed and contains a limit point.
I think that the space $\{\frac{1}{n}\mid n\in\mathbb{N},n>0\}\cup \{0\}$ with the standard Euclidean metric is an answer, but it is not true open ball with center 1 is closed.

$\mathbb{Z}$ with the $p$-adic metric $d(m,n) = p^{-\nu_p (m-n)}$ has the property that every open ball is closed (to prove this, note that the set of positive distances forms a discrete space), and every integer $n$ is the limit point of the sequence $\{n+p^k\}_{k\geq 1}$.
