nLab Cantor space

Idea

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

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

2. removing from these two pieces their middle thirds, to retain the pieces $[0,\tfrac{1}{9}]$, $[\tfrac{2}{9},\tfrac{1}{3}]$, $[\tfrac{2}{3}, \tfrac{7}{9}]$, $[\tfrac{8}{9},1]$;

3. and so ever on.

$$\array{ \mathrlap{[0} && &,& && \mathllap{1]} \\ \mathrlap{[0} &,& \mathllap{\tfrac{1}{3}]} &\phantom{(\tfrac{1}{2}, \tfrac{2}{3})} & \mathrlap{[\tfrac{2}{3}} &,& \mathllap{1]} \\ \mathrlap{[0,\tfrac{1}{9}]} & \phantom{AAAAAAAA} & \mathllap{[\tfrac{2}{9}, \tfrac{1}{3} ]} & \phantom{ (\tfrac{1}{3}, \tfrac{2}{3}) } & \mathrlap{[\tfrac{2}{3}, \tfrac{7}{9}]} & \phantom{AAAAAAAA} & \mathllap{[\tfrac{8}{9}, 1]} \\ &&& \vdots }$$

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 “space-filling curve” (q.v.).

Definition

Traditionally the Cantor space was conceived of

• as a subspace of the real line

but of course it may also be described

• as an abstract space

in itself.

As an abstract space

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

Recall that a binary digit is either $0$ or $1$; the set (or discrete space) of binary digits is the Boolean domain $\mathbb{B}$.

A point in Cantor space is an infinite sequence of binary digits. Accordingly, Cantor space may be denoted $\mathbb{B}^{\mathbb{N}}$, since its set of points is a function set.

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

• If $u \in G$ and $v$ is an extension of $u$ (that is $u$ with possibly additional digits added to the end), then $v \in G$;

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

A point $\alpha$ belongs to an open $G$ if, for some $u$ in $G$, $\alpha$ is an extension of $u$.

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

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 $G$ 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 $G \leq H$ if $G \subseteq H$ as subsets of $\mathbb{B}^*$. 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 $Disc(\{0,1\})$ for the the discrete topological space with two points. Write $\underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})$ 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 $\underset{n \in \mathbb{N}}{\prod} \{0,1\}$ equipped with the Tychonoff topology induced from the discrete topology of $\{0,1\}$).

Then consider the function

$$\array{ \underset{n \in \mathbb{N}}{\prod} &\overset{\kappa}{\longrightarrow}& [0,1] \\ (a_i)_{i \in \mathbb{N}} &\overset{\phantom{AAAA}}{\mapsto}& \underoverset{i = 1}{\infty}{\sum} \frac{2 a_i}{3^i} }$$

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 $\kappa\left( \underset{n \in \mathbb{N}}{\prod} \{0,1\}\right) \subset [0,1]$ equipped with its subspace topology, then this is a homeomorphism onto its image:

$$\underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\}) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} \kappa\left( \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\}) \right) \overset{\phantom{AAAA}}{\hookrightarrow} [0,1] \,.$$

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 $\sim$ in $\mathbb{R}$ (as a binary relation on rational numbers, as described at locale of real numbers), this is mapped to the open $G$ in Cantor space such that $u \in G$ if and only if

$$\sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } \sim \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } + \frac 1 { 3^{-len(u)} } .$$

One then checks that this is an embedding.

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

In either case, the idea is:

• A point of Cantor space corresponds to a number written in base $3$ with infinitely many digits, using only the digits $0$ and $2$ (which are the options for $2 a_i$ when $a_i \in \{0,1\}$); while

• An open corresponds to a union of intervals, each of which is given by approximating a number in base $3$ to a finite number of digits, using only the digits $0$ and $2$.

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 $(\mathbb{Z}/2)^n$ of countably many copies of $\mathbb{Z}/(2)$, which carries a group structure, we can view Cantor space $C$ 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:

• Cantor space is a compact Hausdorff space. (For the topological space, this statement is again equivalent to the fan theorem; for the locale, it holds regardless.)

• Cantor space is totally disconnected.

• Thus Cantor space is a Stone space.

• Cantor space is metrizable, and every compact metrizable space is a quotient space of Cantor space (see Theorem below).

• As a subspace of $\mathbb{R}$, the Cantor set is perfect and uncountable but of Lebesgue measure zero.

• The Cantor set is a precisely self-similar fractal with Hausdorff dimension $\log_3 2 \approx 0.631$.

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 Brouwer‘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 $\widebar{X}$ of a space $X$ that is second-countable locally compact Hausdorff, totally disconnected and perfect, is homeomorphic to Cantor space (provided $X$ is not itself compact).

Proof

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

It follows from this result that all such spaces $X$ 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 $n \geq 1$ of points from Cantor space.

Cantor space is also a “universal” 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 $X$ is separable: has a countable dense set $\{x_0, x_1, \ldots\}$. Assume, as we may, that the metric $d: X \times X \to [0, \infty)$ is valued in $[0, 1]$. Then the map $y: X \to [0, 1]^\mathbb{N}$ to the Hilbert cube, defined by

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

(a type of restricted “Yoneda embedding”, regarding metric spaces as enriched categories), is continuous and maps $X$ onto a closed subspace of $I^\mathbb{N}$. As mentioned at Peano curve, there is a continuous surjection $C \to I^\mathbb{N}$. Taking the pullback in Top

$$\array{ K & \hookrightarrow & C \\ \mathllap{surj} \downarrow & & \downarrow \mathrlap{surj} \\ X & \stackrel{y}{\hookrightarrow} & I^\mathbb{N} }$$

we see that to produce a continuous surjection $C \to X$, it suffices to exhibit a continuous surjection $C \to K$.

In fact, every closed subspace $K \hookrightarrow C$ admits a retraction. There is a clever trick for seeing this: represent Cantor space $C$ instead as the subspace of $[0, 1]$ whose points, when written in base $6$, have just $0$'s and $5$'s in their representation. This subspace has the geometric property that if $x, y \in C$, then $\frac{x+y}{2} \notin C$. As a result, for $x, y, z \in C$ we have $d(x, y) = d(x, z)$ only if $y = z$ and so: given a closed subspace $K$ of $C$, there is for each $x \in C$ a unique element $k_x \in K$ such that $d(x, k_x) = d(x, K)$. The assignment $x \mapsto k_x$ is continuous (in fact a locally constant function on $C \setminus K$, and continuous on $K$ as is easily seen) and provides the desired retraction.

Related concepts

• Peano curve

• Sierpinski space

• long line, Warsaw circle,

• line with two origins

References

Named after Georg Cantor.

• Friedhelm Waldhausen, p. 3-4 in Topologie (pdf)

• Proof Wiki, Cantor Space as Countably Infinite Product