A continuous localization and completion

The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localizationand completion of an An-space is given to be an An-homomorphism between An-spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent G-CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent DCW-complex for a polyhedral categoryD.