Nest generated intersection rings in Tychonoff spaces

Abstract

Tychonoff spaces were characterized by Frink [4] as T1-spaces possessing normal bases. The zero-sets of continuous real-valued functions provide the best known normal base, but in general there are many others. They need not consist entirely of zero-sets. Each normal base leads to a Hausdorff compactification (called Wallman because of the construction used). If the normal base consists entirely of zerosets, the compactification has been called a Z-compactification [15]. This generalization of the Wallman procedure was also considered earlier by Sanin [12] and Banaschewski [2], but apparently Frink was first to consider whether all Hausdorff compactifications might be obtained in this way. Frink was also concerned about the hereditary property of complete regularity and asked for a direct internal proof of this using the normal base characterization. A complication lies in the fact that the trace of a normal base on a subspace need not be a normal base. In [13], one of the authors showed that completely regular spaces (not necessarily T1) are characterized by possessing a stronger base, the trace of which forms a base with the same properties. These bases are called separating nest generated intersection rings. The hereditary property of such rings shows the hereditary property of complete regularity. In this paper we consider the Wallman compactification co(X, Y) and the Wallman realcompactification v(X, 3) associated with a given separating nest generated intersection ring. v(X, Y) is constructed in w(X, 3) in the same way that the Hewitt realcompactification v(X) is constructed in the Stone-Cech compactification :(X). Alo and Shapiro [1] have considered normal bases which are intersection rings (but not necessarily nest generated). Following the analogy with v(X) in :(X), they construct a space -q(X) and show that in many cases it is realcompact. They ask whether this is always the case and also whether all realcompactifications may be obtained in this manner. In ?3 we provide examples which give negative answers to both of these questions. Thus nest generation cannot be relaxed and still keep v(X, 3) realcompact. In ?4 we investigate the relation between separating nest generated intersection rings and certain inverse-closed subalgebras of C(X) (see [6], [7], [9]). Earlier in