 Cantor space

Cantor space is the binary sequence space with the product topology of discrete two-point factors. Here the Bernoulli-space synonym refers to this topology, without selecting a particular probability measure. The metric $d(x,y)={\sum }_{j\ge 1}{2}^{-j}|x_j-y_j|$ induces the topology. A diagonal subsequence argument gives compactness. Fixing a finite prefix gives a clopen cylinder set. The map $x\mapsto {\sum }_{j}2x_j{3}^{-j}$ is a homeomorphism to the usual Cantor set.

Cantor space, often denoted as \(2^{\mathbb{N}}\), is a topological space that is fundamental in various areas of mathematics, particularly in topology and set theory. It is typically constructed as follows: 1. **Definition**: Cantor space consists of all infinite sequences of binary digits (0s and 1s).