nLab
Cantor space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

What is called Cantor space, after Georg Cantor, is the topological space obtained from the closed interval [0,1][0,1] by

  1. removing the middle third, retaining only [0,13][0,\tfrac{1}{3}] and [23,1][\tfrac{2}{3}, 1];

  2. removing from these two pieces their middle thirds, to retain the pieces [0,19][0,\tfrac{1}{9}], [29,13][\tfrac{2}{9},\tfrac{1}{3}], [23,79][\tfrac{2}{3}, \tfrac{7}{9}], [89,1][\tfrac{8}{9},1];

  3. and so ever on.

[0 , 1] [0 , 13] (12,23) [23 , 1] [0,19] AAAAAAAA [29,13] (13,23) [23,79] AAAAAAAA [89,1] โ‹ฎ \array{ \mathrlap{[0} && &,& && \mathllap{1]} \\ \mathrlap{[0} &,& \mathllap{\tfrac{1}{3}]} &\phantom{(\tfrac{1}{2}, \tfrac{2}{3})} & \mathrlap{[\tfrac{2}{3}} &,& \mathllap{1]} \\ \mathrlap{[0,\tfrac{1}{9}]} & \phantom{AAAAAAAA} & \mathllap{[\tfrac{2}{9}, \tfrac{1}{3} ]} & \phantom{ (\tfrac{1}{3}, \tfrac{2}{3}) } & \mathrlap{[\tfrac{2}{3}, \tfrac{7}{9}]} & \phantom{AAAAAAAA} & \mathllap{[\tfrac{8}{9}, 1]} \\ &&& \vdots }

Cantor space is a fundamental object of descriptive set theory; some indications of its use may be found at Polish space. Among its applications is a simple construction of a โ€œspace-filling curveโ€ (q.v.).

Definition

Traditionally the Cantor space was conceived of

but of course it may also be described

in itself.

As an abstract topological space

In brief, Cantor space may be abstractly described as the topological product of countable many copies of the discrete space {0,1}\{0, 1\}. In more concrete detail:

Recall that a binary digit is either 00 or 11; the set (or discrete space) of binary digits is the Boolean domain ๐”น\mathbb{B}.

A point in Cantor space is an infinite sequence of binary digits. Accordingly, Cantor space may be denoted ๐”น โ„•\mathbb{B}^{\mathbb{N}}, since its set of points is a function set.

An open in Cantor space is a collection GG of finite sequences of binary digits (that is a subset of the free monoid ๐”น *\mathbb{B}^*) such that:

  • If uโˆˆGu \in G and vv is an extension of uu (that is uu with possibly additional digits added to the end), then vโˆˆGv \in G;

  • If u:0โˆˆGu:0 \in G and u:1โˆˆGu:1 \in G (where u:iu:i is the immediate extension of uu by the digit ii), then uโˆˆGu \in G.

A point ฮฑ\alpha belongs to an open GG if, for some uu in GG, ฮฑ\alpha is an extension of uu.

An alternative characterization of Cantor space is as the terminal coalgebra for the endofunctor on Top, Xโ†ฆX+XX \mapsto X + X.

What kind of space?

Traditionally, Cantor space is understood as a topological space. We start with the points, as defined above, then specify which sets of points are open. Although there are other ways to state which sets are open, we may define a set to be open if it is the set of points that belong to some open GG as defined above.

A newer approach is to understand Cantor space as a locale. Then we start with the opens and define an order relation on them to define a frame. In this case, the order relation is the obvious one, that Gโ‰คHG \leq H if GโŠ†HG \subseteq H as subsets of ๐”น *\mathbb{B}^*. Then the points come for free, and correspond precisely to the points as defined above.

In classical mathematics, these two approaches are equivalent; a point is determined by its opens, and an open is determined by its points. The theorem that a point is determined by its opens (so that Cantor space, as a topological space, is sober) is valid internal to any pretopos with an exponentiable natural numbers object; as such, it applies even in predicative and constructive mathematics. However, the theorem that an open is determined by its points (so that Cantor space, as a locale, is topological) is equivalent to the fan theorem; it is true in some pretoposes and accepted by some schools of constructivism but false in other pretoposes and rejected, or even refuted, by other constructivists.

When the fan theorem is not valid, the localic approach is probably better; it allows more of the useful properties of Cantor space to hold.

As a subspace of the real line

Cantor space is usually conceived of as a topological subspace of the real line:

Write Disc({0,1})Disc(\{0,1\}) for the the discrete topological space with two points. Write โˆnโˆˆโ„•Disc({0,2})\underset{n \in \mathbb{N}}{\prod} Disc(\{0,2\}) for the product topological space of a countable set of copies of this discrete space with itself (i.e. the corresponding Cartesian product of sets โˆnโˆˆโ„•{0,1}\underset{n \in \mathbb{N}}{\prod} \{0,1\} equipped with the Tychonoff topology induced from the discrete topology of {0,1}\{0,1\}).

Then consider the function

โˆnโˆˆโ„• โŸถฮบ [0,1] (a i) iโˆˆโ„• โ†ฆAAAA โˆ‘i=0โˆž2a i3 i+1 \array{ \underset{n \in \mathbb{N}}{\prod} &\overset{\kappa}{\longrightarrow}& [0,1] \\ (a_i)_{i \in \mathbb{N}} &\overset{\phantom{AAAA}}{\mapsto}& \underoverset{i = 0}{\infty}{\sum} \frac{2 a_i}{ 3^{i+1}} }

which sends an element in the product space, hence a sequence of binary digits, to the value of the power series as shown on the right.

One checks that this is a continuous function (from the product topology to the Euclidean metric topology on the closed interval). Moreover with its image ฮบ(โˆnโˆˆโ„•{0,1})โŠ‚[0,1]\kappa\left( \underset{n \in \mathbb{N}}{\prod} \{0,1\}\right) \subset [0,1] equipped with its subspace topology, then this is a homeomorphism onto its image:

โˆnโˆˆโ„•Disc({0,1})โŸถAAโ‰ƒAAฮบ(โˆnโˆˆโ„•Disc({0,1}))โ†ชAAAA[0,1]. \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\}) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} \kappa\left( \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\}) \right) \overset{\phantom{AAAA}}{\hookrightarrow} [0,1] \,.

This image is the Cantor space as a subspace of the closed interval.

From the localic perspective, a continuous map is given by a homomorphism of frames in the opposite direction. Given an open โˆผ\sim in โ„\mathbb{R} (as a binary relation on rational numbers, as described at locale of real numbers), this is mapped to the open GG in Cantor space such that uโˆˆGu \in G if and only if

โˆ‘ i=1 len(u)2u i3 iโˆผโˆ‘ i=1 len(u)2u i3 i+13 โˆ’len(u). \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } \sim \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } + \frac 1 { 3^{-len(u)} } .

One then checks that this is an embedding.

I should check this some day; for the moment, I am taking it on faith. โ€”Toby

In either case, the idea is:

  • A point of Cantor space corresponds to a number written in base 33 with infinitely many digits, using only the digits 00 and 22; while

  • An open corresponds to a union of intervals, each of which is given by approximating a number in base 33 to a finite number of digits, using only the digits 00 and 22.

One sometimes speaks of the Cantor set to stress that one is considering Cantor space as a subspace of the real line.

As we can also consider Cantor space as a product space (โ„ค/2) n(\mathbb{Z}/2)^n of countably many copies of โ„ค/(2)\mathbb{Z}/(2), which carries a group structure, we can view Cantor space CC as a topological group. In particular, it is a homogeneous space (its group of self-homeomorphisms acts transitively on the space).

Properties

Cantor space, especially in its guise as a subspace of the real line, is quite famous; see Wikipedia. Here are some headline properties:

Theorem

A topological space is homeomorphic to Cantor space if and only if it is nonempty, compact, totally disconnected, metrizable, and perfect.

This result is sometimes called Brouwerโ€™s theorem. It can be seen from the perspective of Stone duality, where the dual result is that any two countable atomless Boolean algebras are isomorphic; this dual result can be proven by a back-and-forth argument.

Corollary

The one-point compactification Xยฏ\widebar{X} of a space XX that is second-countable locally compact Hausdorff, totally disconnected and perfect, is homeomorphic to Cantor space (provided XX is not itself compact).

Proof

Xยฏ\widebar{X} is also second-countable, compact Hausdorff and therefore compact regular, and so by the Urysohn metrization theorem it is compact metrizable. The point pp at infinity is not isolated since we assume XX is not compact, so Xยฏ\widebar{X} is perfect. If VV is any open neighborhood of pp, so V=ยฌKV = \neg K for some compact KโŠ‚XK \subset X, then we claim there exists a clopen that contains KK; in that case VV contains a clopen whence {p}\{p\} is the quasi-component of pp (hence also the connected component since weโ€™re in a compact Hausdorff space). But the argument here shows that for each xโˆˆKx \in K there is a clopen neighborhood of xx contained in ยฌ{p}\neg \{p\}; finitely many of these clopens cover KK, and the claim follows by considering their union.

It follows from this result that all such spaces XX are homeomorphic: they all have Cantor space as their one-point compactifications, and so they are all homeomorphic to the space obtained obtained by removing a single point from Cantor space. This applies for example to spaces obtained by removing a finite number nโ‰ฅ1n \geq 1 of points from Cantor space.

Cantor space is also a โ€œuniversalโ€ compact metric space in the following sense.

Theorem

(Hausdorff-Alexandroff) Every compact metric space is a continuous image of Cantor space.

This implies that every compact metric space is a quotient space of Cantor space, since a surjective map between compact Hausdorff spaces is a closed surjection, and closed surjections are quotient maps.

Proof

First, every compact metric space XX is separable: has a countable dense set {x 0,x 1,โ€ฆ}\{x_0, x_1, \ldots\}. Assume, as we may, that the metric d:Xร—Xโ†’[0,โˆž)d: X \times X \to [0, \infty) is valued in [0,1][0, 1]. Then the map y:Xโ†’[0,1] โ„•y: X \to [0, 1]^\mathbb{N} to the Hilbert cube, defined by

y(x)=(d(x,x n)) nโˆˆโ„•y(x) = (d(x, x_n))_{n \in \mathbb{N}}

(a type of restricted โ€œYoneda embeddingโ€, regarding metric spaces as enriched categories), is continuous and maps XX onto a closed subspace of I โ„•I^\mathbb{N}. As mentioned at Peano curve, there is a continuous surjection Cโ†’I โ„•C \to I^\mathbb{N}. Taking the pullback in Top

K โ†ช C surjโ†“ โ†“surj X โ†ชy I โ„•\array{ K & \hookrightarrow & C \\ \mathllap{surj} \downarrow & & \downarrow \mathrlap{surj} \\ X & \stackrel{y}{\hookrightarrow} & I^\mathbb{N} }

we see that to produce a continuous surjection Cโ†’XC \to X, it suffices to exhibit a continuous surjection Cโ†’KC \to K.

In fact, every closed subspace Kโ†ชCK \hookrightarrow C admits a retraction. There is a clever trick for seeing this: represent Cantor space CC instead as the subspace of [0,1][0, 1] whose points, when written in base 66, have just 00's and 55's in their representation. This subspace has the geometric property that if x,yโˆˆCx, y \in C, then x+y2โˆ‰C\frac{x+y}{2} \notin C. As a result, for x,y,zโˆˆCx, y, z \in C we have d(x,y)=d(x,z)d(x, y) = d(x, z) only if y=zy = z and so: given a closed subspace KK of CC, there is for each xโˆˆCx \in C a unique element k xโˆˆKk_x \in K such that d(x,k x)=d(x,K)d(x, k_x) = d(x, K). The assignment xโ†ฆk xx \mapsto k_x is continuous (in fact a locally constant function on Cโˆ–KC \setminus K, and continuous on KK as is easily seen) and provides the desired retraction.

References

Revised on May 16, 2017 08:19:40 by Urs Schreiber (131.220.184.222)