Homotopy Limits, Completions and Localizations

The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan| for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others.