Remarks on the regularity of boundary points in a resolutive compactification
Teruo Ikegami · Osaka City University (Osaka City University) · 1980
Introduction.Let X be a strong harmonic space in the sense of Bauer [2] and suppose that constant functions are harmonic.In the previous paper [5], the author studied the regularity of boundary points in a resolutive compactification of X and discussed characterization of regularity, existence of regular points, strong regularity and pseudo-sΐrong regularity, characterization of harmonic boundary and consideration in the case of open subsets.In this paper we shall use the same notations and definitions as in [5], and we shall give some supplementary remarks.In §1, we recall the notations and terminologies used in [5].We reform characterization of the regularity in Theorem 1 of §2.Theorem 2 in §3 is the extremal characterization of pseudo-strong regularity in the case where X is a Brelot space.The trace filters of neighborhoods of boundary points in the Wiener compactification X w of X is of some interes:.Using this filters we can construct in §4 a family of completely regular filters in a metrizable and resolutive compactification X* of X.A regular boundary point x is said to have a local property if x is regular for every U(x) Π X, where U(x) is a neighborhood of x.The main results of this paper are in §5.It is shown that a regular point x does not possess a local property in general and x has a local property if and only if x is pseudo-strongly regular.Further the related problems are investigated.In the final section, we consider a relatively compact open set G of a Brelot space and obtain the result, if G is minimally bounded, then the set of all regular points is dense in the boundary dG of G, which is a generalization of a result of Bauer [1].