Remainders of H-closed extensions

AbstractThe remainders of compactifications of Tychonoff spaces have been studied extensively during the last five decades, but, the remainders of H-closed extensions of Hausdorff spaces have not been studied in a concerted manner so far. In this paper, we show that two Hausdorff spaces X and Y have the same remainders in their H-closed extensions if there exists a perfect, irreducible and θ-continuous mapping from X onto Y. Further, a Hausdorff space Z is shown to be the remainder of some H-closed extension of a given space X if and only if there exists a perfect (not necessarily continuous) mapping from σX⧹X onto Z. We also show that if X is locally H-closed and |σX⧹X|⩾ℵ0, then the set of remainders of H-closed extensions of X contains all separable H-closed spaces. We give examples to illustrate the differences between the sets of remainders of compactifications and remainders of H-closed extensions of a fixed Tychonoff space