TITLE: What is a reference for profinite sets?
QUESTION [8 upvotes]: 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.

REPLY [6 votes]: 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.