of generators that behave a certain way; alternatively, one might start with a set and ask which algebra it generates. Every set X generates a free Boolean algebra FX defined as the algebra such that for every algebra B and function f : X → B, there is a unique Boolean algebra homomorphism f′ : FX → B that extends f. Diagrammatically, where iX is the inclusion, and the dashed arrow denotes uniqueness. The idea is that once one chooses where to send the elements of X, the laws for Boolean algebra homomorphisms determine where to send everything else in the free algebra FX. If FX contained elements inexpressible as combinations of elements of X, then f′ wouldn't be unique, and if the elements of X weren't sufficiently independent, then f′ wouldn't be well defined! It is easily shown that FX is unique (up to isomorphism), so this definition makes sense. It is also easily shown that a free Boolean algebra with generating set X, as defined originally, is isomorphic to FX, so the two definitions agree. One shortcoming of the above definition is that the diagram doesn't capture that f′ is a homomorphism; since it is a diagram in Set each arrow denotes a mere function. We can fix this by separating it into two diagrams, one in BA and one in Set. To relate the two, we introduce a functor U : BA → Set that "forgets" the algebraic structure, mapping algebras and homomorphisms to their underlying sets and functions. If we interpret the top arrow as a diagram in BA and the bottom triangle as a diagram in Set, then this diagram properly expresses that every function f : X → UB extends to a unique Boolean algebra homomorphism f′ : FX → B. The functor U can be thought of as a device to pull the homomorphism f′ back into Set so it can be related to f. The remarkable aspect of this is that the latter diagram is one of the various (equivalent) definitions of when two functors are adjoint. Our F easily extends to a functor Set → BA, and our definition of X generating a free Boolean algebra FX is precisely that U has a left adjoint F. 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