General topology Cantor sets and descriptive spaces Example

Cantor set

Example

Let $C_0 = [0, 1]$ . For $n \ge 0$ , let $C_{n+1}$ be the set that we get from $C_n$ when we remove the open middle third of each of its closed intervals. Thus $C_1 = [0, \tfrac{1}{3}] \cup [\tfrac{2}{3}, 1]$ . The Cantor set is $C = \bigcap_{n \ge 0} C_n$ , with the subspace topology of $\mathbb{R}$ .

Remarks

Equivalently, $C$ is the set of $x \in [0, 1]$ that have a base-3 expansion $x = \sum_{n \ge 1} a_n 3^{-n}$ with all $a_n \in \{0, 2\}$ . The set $C$ is closed and bounded, so it is compact. It is totally disconnected and has no isolated points. It is uncountable, and it is homeomorphic to the space of binary sequences. Its Lebesgue measure is $0$ , because $C_n$ has total length $(2/3)^n$ . The Cantor set is nowhere dense in $\mathbb{R}$ .

Tag	`0BXNDZFQME`
URI	`m2://0BXNDZFQME`
Aliases	`cantor-ternary-set`
Kind	example
Writing	W2 usable
MSC	54F65
Instance of	Compact space, Totally disconnected space
Same as	Wikidata Q273188 Wikipedia Cantor set

Cantor set

Example

Remarks

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