nLab
Cantor space

Idea

Cantor space, named after Georg Cantor, is a famous space. Cantor studied it primarily as a subspace of the real line, but it is also important as a space in its own right.

Definition

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

A point in Cantor space is an infinite sequence of binary digits. Accordingly, Cantor space may be denoted ${𝔹}^{\mathbb{N}}$, 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 ${𝔹}^{*}$) 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 $u$.

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 $G\le H$ if $G\subseteq H$ as subsets of ${𝔹}^{*}$. 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

Cantor space is usually conceived of as a subspace of the real line. Pointwise, it is easy to define the embedding from ${𝔹}^{\mathbb{N}}$ into $ℝ$; we map the infinite sequence $\alpha$ to the real number

$$\sum _{i=1}^{\mathrm{\infty }}\frac{2{\alpha }_i}{3^i}.$$\sum_{i=1}^{\infty} \frac { 2 \alpha_i } { 3^i } .

One then checks that this function is in fact an embedding.

From the localic perspective, a continuous map is given by a homomorphism of frames in the opposite direction. Given an open $\sim$ in $ℝ$ (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

$$\sum _{i=1}^{\mathrm{len}(u)}\frac{2u_i}{3^i}\sim \sum _{i=1}^{\mathrm{len}(u)}\frac{2u_i}{3^i}+\frac{1}{3^{-\mathrm{len}(u)}}.$$\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. —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 $2$; 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.

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.

• Every compact metric space is (as a topological space) a quotient space of Cantor space.

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

• The Cantor set is a precisely self-similar fractal? with Hausdorff dimension ${\log }_{3}2\approx 0.631$.