Topological Realization
The free Boolean algebra with κ generators, where κ is a finite or infinite cardinal number, may be realized as the collection of all clopen subsets of {0,1}κ, given the product topology assuming that {0,1} has the discrete topology. For each α<κ, the αth generator is the set of all elements of {0,1}κ whose αth coordinate is 1. In particular, the free Boolean algebra with generators is the collection of all clopen subsets of a Cantor space. Surprisingly, this collection is countable. In fact, while the free Boolean algebra with n generators, n finite, has cardinality, the free Boolean algebra with generators has cardinality .
For more on this topological approach to free Boolean algebra, see Stone's representation theorem for Boolean algebras.
Read more about this topic: Free Boolean Algebra
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