What is a reference for profinite sets? The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not exactly a wealth of references about profinite sets and profinite completions of sets.
 A: Profinite sets are just another name for compact totally disconnected topological spaces.  I think this is (essentially) explained somewhere in Bourbaki's books on general topology.
A: I agree with some of the comments that "profinite set" is not a standard term.  But you can certainly look at the category of pro-(finite sets).  In other words, begin with the category $Set_f$ of finite sets and functions.  Then one can form a category $Pro(Set_f)$ as the projective completion of the category $Set_f$; it is the full subcategory of the category of functors from $Set_f$ to $Set$, consisting of objects isomorphic to projective limits of systems of finite sets.  In other words, the objects of $Pro(Set_f)$ are (not necessarily representable) functors from $Set_f$ to $Set$, which are inductive (viewing finite sets via Yoneda as functors from $Set_f$ to $Set$ switches arrow directions) limits of representable functors from $Set_f$ to $Set$.  I'm sure one should be careful about some smallness/universe issues to make this precise.
The category $Pro(Set_f)$ is equivalent to the category of compact totally disconnected topological spaces.  This elaborates on Leonid's answer.
The reference for this somewhat highbrow answer is a paper of Gaitsgory and Kazhdan, in GAFA, titled "Representations of algebraic groups over a 2-dimensional local field".  
A: Johnstone's book Stone Spaces is a suitable reference for this and much more. The book by Ribes and Zalesskii Profinite Groups also has good information.  Algebre et Théories Galoisienne by Douady and Douady also gives a nice accounting.
A: See http://math.harvard.edu/~waffle/boolean.pdf for notes on Boolean algebras and the fact that they are essentially (anti-)equivalent to totally disconnected compact Hausdorff spaces.
