### 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 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*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.#### Keywords

n-Hausdorff spaces; H-closed spaces; Shapiro's absolute

#### Full Text:

PDFDOI: https://doi.org/10.1478/AAPP.992A1

Copyright (c) 2021 Fortunata Aurora Basile, Maddalena Bonanzinga, Nathan Carlson, Jack Porter

This work is licensed under a Creative Commons Attribution 4.0 International License.