Cantor algebra

Example

Let $C = \set{0, 1}^{\N}$ be the Cantor space of infinite binary sequences, with the product topology. The countable Cantor algebra is the Boolean algebra of all clopen subsets of $C$, with union, intersection and complement in $C$.

For a finite binary word $w$, let $[w]$ be the set of sequences that begin with $w$. Each set $[w]$ is clopen, and every clopen subset of $C$ is a finite union of such sets, by compactness of $C$.

Check the main properties.

1. It is countable. There are countably many finite words, hence countably many finite unions of sets $[w]$.
2. It has no atoms. A nonempty clopen set contains some $[w]$. The set $[w]$ is the disjoint union of the nonempty sets $[w0]$ and $[w1]$, so it is not minimal.
3. It is free on the generators $G_n = \set{s \in C : s_n = 1}$ for $n \in \N$. Each $[w]$ is an intersection of finitely many sets $G_n$ and complements of sets $G_n$, so these sets generate the algebra. Any such intersection over distinct indices $n$ is nonempty. So no nontrivial Boolean relation holds among the $G_n$.