Cantor set is closed

Statement

The Cantor set $C = \bigcap_{n \ge 0} C_n$ is a closed subset 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. Since $C \subseteq [0, 1]$ , the set $C$ is also bounded, and thus compact by the Heine–Borel theorem. Each endpoint of an interval of some $C_n$ lies in $C$ . For example, $\tfrac{1}{3}$ and $\tfrac{2}{3}$ are in $C$ .

Tag	`01VT24J1SD`
URI	`m2://01VT24J1SD`
Kind	statement, proposition
Writing	W0 stub
MSC	54F65
Status	This copy of the site has no build data. Run `m2 status 01VT24J1SD`.
Uses	Cantor set, Intersection of closed sets is closed, Closed set

Tag

01VT24J1SD

URI

m2://01VT24J1SD

Kind

statement, proposition

Writing

W0 stub

MSC

54F65

Status

This copy of the site has no build data. Run m2 status 01VT24J1SD.

Uses

Cantor set, Intersection of closed sets is closed, Closed set

Cantor set is closed

Statement

Remarks

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