Round subsets of Wallman-type compactifications

Abstract

Let 2T be a normal base of a Tychonoff space X and w(3t) (v(3C)) denote the Wallman-type (real-) compactification of X generated by 3t. This Wallman-type compactification is known to associate with a unique proximity S. A 3. -filter &• is round if for each F £^ there is a n F G £^ such that F 0 E(X -F). A subset A of (o(3t) is called a round subset of <u(2T) iff for each Z e 2, if C l^Z contains A, then it is a neighborhood of A. Properties of round 3?-filters and round sets of a>(2?) are introduced. We also prove that the intersection of all the free 2T-ultrafilters is 9 = {Z e Z: C\ X (X -Z) is compact} iff o>(S)-X is a round subset of &>(2T); if 3T is a separating nest generated intersection ring with property (a) then w(S)-v(3?) is a round subset of u>(2).