Show that there is a metric space that has a limit point, and each open disk in it is closed. Collecting examples

Question

Show that there is a metric space that has a limit point, and each open disk in it is closed.

This question belongs to the 39th math competitions of Iran. This is one solution:

Suppose that $X=\{\frac{1}{n}: n\in \mathbb{N}\} \cup \{0\}$ and:

$d(x,y) = \left\{ \begin{array}{ll} x+y & \mbox{if } x \neq y \\ 0 & \mbox{if } x = y \end{array} \right.$

It is clear that $(X,d)$ is a metric space and $0$ is a limit point for this space. And for every $x\in X$ and $r > 0$ the open disk $B_r(x)$ has one element or for one $1\leq N$ its equal to $\{\frac{1}{n}: n\geq N\} \cup \{0,x\}$. In each case they are closed.

I am looking for other solutions for this question.

Answer 1

The p-adic numbers are an example of this phenomenon and there are a lot of similar examples, since any discrete valuation ring or its quotient field have the property that open disks are closed and vice versa. This includes finite extensions of the p-adics as well as rings of the form $K((X))$ (Laurent series with finite principal part over a field).

Note that these are more or less algebraic examples which might not be what you are looking for since the question is tagged as real-analytic.