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, sometimes called the Cantor algebra. This collection is countable. In fact, while the free Boolean algebra with n generators, n finite, has cardinality , the free Boolean algebra with generators, as for any free algebra with generators and countably many finitary operations, has cardinality . For more on this topological approach to free Boolean algebra, see Stone's representation theorem for Boolean algebras. See also Boolean algebra (structure) Generating set References Steve Awodey (2006) Category Theory (Oxford Logic Guides 49). Oxford University Press. Paul Halmos and Steven Givant (1998) Logic as Algebra. Mathematical Association of America. Saunders Mac Lane (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag. Saunders Mac Lane (1999) Algebra, 3d. ed. American Mathematical Society. . Robert R. Stoll, 1963. Set Theory and Logic, chpt. 6.7. Dover reprint 1979. Boolean algebra Free algebraic structures