---
tag: 415776K9Z3
kind: definition
name: Perfect set property
msc: ["03E15", "54H05"]
defines: H47ZGRR13V
uses: [584GXXY73M, 628242KN0T, 01TQB88H4C]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_set_property
---

## Definition

Let $X$ be a [[Polish space|584GXXY73M]] and let $A \subseteq X$.
The set $A$ has the *perfect set property* if $A$ is [[countable|01TQB88H4C]] or $A$ contains a nonempty [[perfect set|628242KN0T]].
Here a perfect set is a closed subset of $X$ with no isolated points.
A class $\Gamma$ of subsets of Polish spaces has the perfect set property if every set in $\Gamma$ has it.

## 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.