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 Błaszczyk (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-Čech compactification) Ŷ is constructed for the semiregularization PX(s) of the absolute PX such that Ŷ is a compact, extremally disconnected, completely regular (but not necessarily Hausdorff) extension of PX(s), and PX(s) is C*-embedded in Ŷ. The definition of the Fomin extension σX for a Hausdorff space X (Porter and Woods 1988) is extended to an arbitrary space X and σX\X is shown to be homeomorphic to the remainder Ŷ\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.