Topology/Cantor Space

The Cantor space is a generalisation of the Cantor set, which can be seen as any element as the homeomorphism class of the Cantor set defined below.

Definition of the Cantor set

The Cantor set is the limit of an infinite process of removing subsets of the unit interval depicted here.

So at any given step we remove the middle thirds of the unit interval, let $C_{0}=[0,1]$ so at the n-th step we have:

$C_{n}={\frac {C_{n-1}}{3}}\cup {\big (}{\frac {2}{3}}+{\frac {C_{n-1}}{3}}{\big )}$

So the effect is scaling the previous step down by a third and putting two copies next to each other. Then the Cantor set is

$C=\bigcap _{i=0}^{\infty }C_{i}$

Definition of the Cantor space

A Cantor space is any space homeomorphically equivalent to $C$ defined above. The value of this definition is elucidated by the following theorem.

Brouwer's Theorem

An equivalent form of Brouwer's original theorem is: "A topological space is a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected, and metrizable."

To understand this let's fill in the definition not yet covered.

1. Clearly, non-empty means it is not the empty set.

2. Perfect means the space has no isolated points (no points whose singleton is an open set in the ambient space).

3. Compactness has been covered.

4. Totally disconnected means the connected components (the maximal connected subsets) of the space are all points.

5. Metrisable means there exists a metric for this space.

Thus the theorem states that the combination of these conditions leaves a space homeomorphic to the Cantor set.

Example

Consider the map from the set of infinite binary codes $\{0,1\}^{\mathbb {N} }$ often written $2^{\mathbb {N} }$ to the Cantor set defined above: $f:2^{\mathbb {N} }\to C$ defined as for a sequence $a=(a_{i})_{i=0}^{\infty }$

$f(a)=\sum _{i=0}^{\infty }{\frac {2a_{i}}{3^{i+1}}}$

Then f is a homeomorphism of metric spaces, meaning that all the properties above hold for the space $2^{\mathbb {N} }$.