Free Boolean algebra

In mathematics, a free Boolean algebra is a free object in the category of Boolean algebras. That is, a Boolean algebra B is free if there is a subset g of B, called the set of generators of B, such that any map from g into a Boolean algebra C extends uniquely to a homomorphism from B into C .

For any desired number κ of generators, finite or infinite, the free Boolean algebra with κ generators may be constructed as the collection of all clopen subsets of {0,1}^κ, given the product topology assuming that {0,1} has the discrete topology. The generators may be enumerated as follows: for each α<κ the α'th generator is the set of all elements of {0,1}^κ whose α'th coordinate is 1. In particular, the free Boolean algebra with ℵ₀ generators is the collection of all clopen subsets of Cantor space. Perhaps surprisingly, there are only countably many of these. In fact, while for finite n the free Boolean algebra with n generators has cardinality $2^{2^{n}}$, for infinite κ the corresponding cardinality is just κ.