---
tag: 01VT24J1SD
kind: statement
role: proposition
name: Cantor set is closed
msc: [54F65]
uses: [0BXNDZFQME, 09YMV8H4X5, 0G5AV9Q4RR]
sources:
  - url: https://en.wikipedia.org/wiki/Cantor_set
    locator: Topological and analytical properties
---

## Statement

The [[Cantor set|0BXNDZFQME]] $C = \bigcap_{n \ge 0} C_n$ is a [[closed subset|0G5AV9Q4RR]] of $\mathbb{R}$.

## Remarks

Proof: each $C_n$ is a union of $2^n$ closed intervals, so it is closed. An [[intersection of closed sets is closed|09YMV8H4X5]].
Since $C \subseteq [0, 1]$, the set $C$ is also bounded, and thus compact by the [[Heine–Borel theorem|07PFYV65MB]].
Each endpoint of an interval of some $C_n$ lies in $C$. For example, $\tfrac{1}{3}$ and $\tfrac{2}{3}$ are in $C$.

## Explanations

### L1 human

At each step we keep a finite union of closed intervals, which is a closed set.
Closed sets stay closed under intersections, so the limit set is closed.