Perfect set

Remarks

The two forms agree because $P$ is closed if and only if $P' \subseteq P$, and $P$ has no isolated points if and only if $P \subseteq P'$.

Example: $[0, 1]$ is closed in $\R$, and every $x \in [0, 1]$ is a limit of other points of $[0, 1]$, so $[0, 1]$ is perfect. The Cantor set is also perfect.

Non-example: $P = \set{0} \cup \set{1/n : n \ge 1}$ is closed in $\R$, but $1$ is an isolated point of $P$, so $P$ is not perfect.

Non-example: $\Q$ has no isolated points, but it is not closed in $\R$, so it is not perfect.

Every nonempty perfect subset of $\R$ is uncountable.