@DanielFischer $\overline{A}$ is the smallest closed set that contains A

$\overline{A}= \bigcap \{ K \subset X: K \text{ closed and } A \subset K\}$

A is closed $\Leftrightarrow$ for each $(x_n) \subset A $ with $x_n \overset{\rho}{\to} x, x \in X$ then $x \in A$. @DanielFischer

@DanielFischer $B_{\rho}(x, \epsilon)=\{ y \in X: \rho(x,y)< \epsilon\}$
U is open if for each $x \in U$ there is a $\epsilon>0$ such that $B_{\rho}(x, \epsilon) \subset U$.

Indeed - I guess we all hope that.
I have a photo of Jonas and I in Amsterdsam, if you would like a copy

@DanielFischer If we apply the definition we get this: $X \setminus{\overline{A}}$ is open , so for each $x \in X \setminus{\overline{A}}$ there is an $\epsilon>0$ such that $B_{\rho}(x, \epsilon) \subset X \setminus{\overline{A}} \Leftrightarrow \{ y \in X: \rho(x,y)< \epsilon\} \subset X \setminus{\overline{A}}$

Can we continue in order to deduce whether $x$ belongs to $X \setminus \overline{A}$ or not?

This is a shot in the dark, but by any chance does this sequence look familiar to anyone here? :0
\begin{align*}
b_{2} &= 1 \\
b_{4} &= 4k-3 \\
b_{6} &= 16k^{2}-39k+20 \\
b_{8} &= 64k^{3} -326k^{2}+492k-210
\end{align*}