Does there exist a complete metric space with (some) disconnected balls and such that $\overline{B(x,r)}=E(x,r)$? In a metric space, let's denote with $B(x,r)$ the open ball of center $x$ and radius $r$ and with $E(x,r)$ the closed ball. While waiting for an official term, let's call naïf a metric space such that $\overline{B(x,r)}=E(x,r)$ holds for all $r>0$ and $x\in X$.
I was answering this question: apparently, there is a theorem according to which in a compact naïf metric space (open) balls are always connected.
When it came to providing examples of non-compact naïf metric spaces with a disconnected open ball, all I could think of was $\Bbb Q$. However, I was wondering

Is there a naïf complete metric space with a disconnected open ball?

Sneaky bonus question: is there a naïf complete metric space with a disconnected closed ball? $\ddot\smile$
As a preliminary observation, by virtue of that theorem the counterexample-ball cannot be relatively compact. Therefore, closed subspaces of $\Bbb R^n$ with the induced metric won't make good counterexamples.
Looking for spaces where no closed ball is compact - say, an appropriate closed subset of $\ell^2$ or $L^\infty$ - seems the most natural choice, but nothing is occurring to me right now.

Warning: For the sake of clarity, the following conditions are deceivingly similar to naïveté on paper, but they are not the same thing: 


*

*"Every open ball is closed": balls that are closed sets are not necessarily closed balls. In fact, the only naïf metric space where this condition holds is the singleton, because we'd need $B(x,r)=\overline{B(x,r)}=E(x,r)$, which always fails for $r=d(x,y)$.

*"The closure of any open ball is a closed ball": same center and radius are needed. $\Bbb N$ with the standard distance satisfies this, but it is not naïf because $\overline{B(x,n)}\ne E(x,n)$ for all positive integers $n$.
However: This criterion describes an equivalent condition to naïveté: "A metric space is naïf if and only if for all $r>0$ and for all $x\ne y\in X$ there is $z\in X$ such that $d(y,z)<\varepsilon$ and $d(x,y)>d(x,z)$".

I think that the complete Erdös space $E_c$ is a candidate. This consists of all sequences $(x_n) \in \ell_2$ such that $x_n \in \{0\} \cup \{ \frac{1}{k}: k \in \mathbb{N}^+ \}$ for all $n$, in the subspace topology.
This is complete one-dimensional and totally disconnected. I think that is "naïf" too (please check).
