---
tag: 628242KN0T
kind: concept
name: Perfect set
msc: [54A05, 54H05]
specializes: [0ZCK8DQYN1]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_set
same_as:
  wikidata: Q7168101
  wikipedia: Perfect set
---

## Summary

A perfect set in a [[topological space|0CQHFPGJ45]] is a [[closed set|0ZCK8DQYN1]] without [[isolated points|0A1YXSKB48]].
Equivalently, it equals its own [[derived set|8ZAPAPYJFG]].
In the real line, every nonempty perfect set is uncountable, and the [[Cantor set|0BXNDZFQME]] is the standard example.

## Explanations

### L1 human

A perfect set contains all the points it accumulates at, and none of its points stands alone.
Every point of it has other points of the set crowding around it.
The closed interval $[0, 1]$ is perfect.
The Cantor set, obtained by removing middle thirds again and again, is perfect too, although it contains no interval at all.