Studying analysis we were given the definition of closed set and accumulation points.
We than proved that given a metric space $(X,d)$, given $A \subseteq X$, $A$ is closed iff $A' \subseteq A$. I was wondering how can $A'$ be a subset and not always equal to $A$, so I did some research and for $\mathbb{R}$ they are in fact equal. Does this extends to arbitrary sets and metrics? If not could you provide some examples where this fails?
In the usual topology on $\Bbb R$, $\{1\}'=\emptyset$ and $[0,1] \cup \{2\} = [0,1]$ so it can be a subset of $A$.