Let $p_1,p_2,p_3,\ldots$ be primitive propositions, and consider the Boolean algebra made up of all propositions you can form from finitely many of them connected by "and", "or", and "not". Each such proposition corresponds to a clopen subset of the Cantor set. If $p_n$ is true, that corresponds to the $n$th ternary digit being $1$; if false then $0$. So a proposition specifies something about finitely many binary digits. Thus a clopen set.
Now let $q_1,q_2,q_3,\ldots$ be more primitive propositions.
The whole set $p_1,p_2,p_3,\ldots,q_1,q_2,q_3,\ldots$ is still countable, so the Boolean algebra you get is isomorphic to the one you get with just the "$p$"s and not the "$q$"s. And it's the product space.
Notice that the union of the two sets of propositions corresponds to the product of two topological spaces. That's one aspect of the "anti-equivalence" of the category of Boolean algebras and Boolean homomorphisms, and the category of totally disconnected compact Hausdorf spaces and continuous functions.