Cantor set is homeomorphic to the space of binary sequences

Builds on Cantor set Product topology on an arbitrary product Continuous bijection from a compact space to a Hausdorff space is a homeomorphism Tychonoff's theorem

Statement

Let $\{0, 1\}$ have the discrete topology and let $\{0, 1\}^{\mathbb{N}}$ have the product topology. Then the map $\varphi \colon \{0, 1\}^{\mathbb{N}} \to C, \qquad \varphi((a_n)_{n \ge 1}) = \sum_{n \ge 1} \frac{2 a_n}{3^n}$ is a homeomorphism onto the Cantor set $C$ .

Remarks

The map is a bijection, because each point of $C$ has exactly one base-3 expansion with digits in $\{0, 2\}$ . It is continuous: if two sequences agree in their first $N$ terms, their images differ by at most $3^{-N}$ . The space $\{0, 1\}^{\mathbb{N}}$ is compact by Tychonoff's theorem, and $C$ is Hausdorff. Thus $\varphi$ is a homeomorphism, by this theorem. A space homeomorphic to $C$ is a Cantor space. By a theorem of Brouwer, these are exactly the nonempty compact metrizable spaces that are totally disconnected and have no isolated points.

Reading path

To understand this page from the ground up, read these 58 pages in order. The foundations come first.

Cantor set example
Element definition
Proposition concept
Real numbers object
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Intersection of a family of sets definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Complement definition
Function definition
Subspace topology definition
Basis of a topology definition
Codomain definition
Injective function definition
Preimage of a set definition
Metric space definition
Product topology definition
Subbasis definition
Surjective function definition
Product topology on an arbitrary product definition
Open ball definition
Bijective function definition
Metric topology definition
Inverse function definition
Open set definition
Closed set definition
Hausdorff space definition
Continuous map of topological spaces definition
Open cover definition
Complement of an open set is closed statement
Homeomorphism definition
Compact space definition
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Closed subsets of compact sets are compact statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Product of compact sets is compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Tychonoff's theorem statement
Cantor set is homeomorphic to the space of binary sequences statement

Where this leads

4 pages build on this one.

Tag	`0RQ8DHHR8V`
URI	`m2://0RQ8DHHR8V`
Kind	statement, theorem
Writing	W0 stub
MSC	54F65
Status	This copy of the site has no build data. Run `m2 status 0RQ8DHHR8V`.
Uses	Cantor set, Product topology on an arbitrary product, Continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Tychonoff's theorem

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Cantor space