---
tag: 4S1HC0N5E8
kind: definition
name: Perfect set
msc: [54A05, 54H05]
defines: 628242KN0T
uses: [0CQHFPGJ45, 0ZCK8DQYN1, 0A1YXSKB48, 8ZAPAPYJFG]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_set
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $P \subseteq X$ is *perfect* if $P$ is [[closed|0ZCK8DQYN1]] in $X$ and $P$ has no [[isolated points|0A1YXSKB48]].
Equivalently, $P = P'$, where $P'$ is the [[derived set|8ZAPAPYJFG]] of $P$.

## 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|0BXNDZFQME]] 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.