The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.
[1] The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generated by a countable number of elements.
As the size of countably generated complete Boolean algebras is unbounded, this shows that there is no free complete Boolean algebra on a countable number of elements.
If κ and λ are cardinals, then the Boolean algebra of regular open sets of the product space κλ is a collapsing algebra.
The simplest option is to take the usual product topology.