Perfect set property

Remarks

Example: the Cantor set $C \subseteq \R$ has the property, because it is nonempty, closed and has no isolated points, so it contains the perfect set $C$ itself.

Non-example: under the axiom of choice there is a Bernstein set $B \subseteq \R$, a set such that both $B$ and $\R \setminus B$ meet every uncountable closed set. Then $B$ is uncountable, but $B$ contains no nonempty perfect set $P$, because $P$ is an uncountable closed set and so meets $\R \setminus B$.

Every closed set has the property by the Cantor–Bendixson theorem, and so does every Borel set and every analytic set.