Absolutes and n -H-closed spaces

Abstract

In this paper the investigation of n -H-closed spaces that was started by Basile et al . (2019) is continued for every n ∈ ω, n ≥ 2. In particular, starting with the relationship between the absolute space PX of an arbitrary topological space X , reported by Ponomarev and Shapiro (1976) and introduced by Blaszczyk (1975, 1977), Ul'yanov (1975a,b) and Shapiro (1976), it is shown that the absolute PX is n -H-closed if and only if X is n -H-closed. For an arbitrary space X , a β-like extension (β for the Stone-Cech compactification) Y is constructed for the semiregularization PX(s) of the absolute PX such that Y is a compact, extremally disconnected, completely regular (but not necessarily Hausdorff) extension of PX(s) , and PX(s) is C* -embedded in Y . The definition of the Fomin extension σ X for a Hausdorff space X (Porter and Woods 1988) is extended to an arbitrary space X and σ X \ X is shown to be homeomorphic to the remainder Y \ PX(s) . A similar result is established when X is an n -Hausdorff space defined by Basile et al. (2019). Further, we give a cardinality bound for any n -Hausdorff space X and show that the inequality |X| ≤ 2^χ( X ) for an H-closed space X proved by Dow and Porter (1982) can be extended to n -H-closed spaces.