I appologize about the terrible formatting job.

What I am calling the profinite completion of a set is almost the same as for a group, except everytime the word group is used, you replace it the with the word set. Now the Stone Cech compactifaication of a (discrete) space misses the words, "totally disconnected". Wikipedia states (so a grain of salt) that the stone cech compactification happens to be totally disconnected. So ths notion of profinite completion seems to be the stone cech compactification. I would like to make a remark on the terminology. Any concrete category with filtered limits has a "profinite completion" functor.

The projective limit definition would amount to the same thing.

Let us first say what a profinite set is. This is a compact Haussdorf totally disconnected topological space. We may form the category of profinite spaces where the morphisms are continuous maps between them. Their is a forgetfull functor from profinite sets to sets that forgets the topology. Profinite completion is the left adjoint to this functor.