A <subset> $F\subseteq X$ is a <closed set> in $X$ when its <complement> is an <open set> in the <metric topology>. Equivalently,
$$
\boxed{x\in X\setminus F\ \Longrightarrow\ \exists r>0:\ B(x,r)\cap F=\varnothing}.
$$
The <open ball> is taken inside the ambient <metric space> $X$; closedness always depends on that ambient space.

Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2023 / 1 / a / ii / Solution

 Solution (source code)