Subsets of C∗(X) generating compactifications

AbstractThe present paper is a continuation of three papers written by B.J. Ball and Shoji Yokura which were concerned with compactifications determined by subsets of C∗(X) and functional bases for subsets of C∗(X).The purpose of this paper is to characterize those subsets of C∗(X) which generate compactifications of X. A number of necessary and sufficient conditions for two subsets of C∗(X) to generate the same compactification are given. A certain assertion is shown to be equivalent to the continuum hypothesis. Another assertion is shown to be equivalent to the condition: nℵ0 = n whenever cf(n) > ℵ0