Weight of a compactification and generating sets of functions

Given a Tychonoff space X, one of the standard methods for construction and comparison of compactifications of X consists in embedding X in real cubes, through subfamilies of C*(X), the ring of bounded continuous real functions on X. We say that a subfamily F of C*(X) generates a compactification aX when the diagonal D(F) embeds X in a cube and the closure of the image is equivalent to aX. We say that F determines aX if aX is the smallest compactification to which every member of F extends. In this paper it is proved that a subset F of C*(X) generates a compactification if and only if the subring generated by F separates points from closed sets. Moreover, if two subsets F and G of C*(X) generate the same subring or have the same closure with respect to uniform convergence topology, then whenever F generates a compactification aX, G also does. This paper also deals with cardinality of sets of functions which determine a compactification. Finally, lattice properties of the set of compactificatons of X having the same weight as X are studied.