Extensions of hausdorff spaces

Abstract

A topological property and is called a Hausdorff extension property if each Hausdorff space X can be densely embedded in a space Î and X such that (1) Î and X is a Hausdorff space with property P (2) if Y is a Hausdorff extension of X with P then there is a continuous function ƒ = K X → Y such that f(x) = x for each x ϵ X, and (3) if X is H-closed, then X=X. Both necessary conditions and sufficient conditions are given to characterize Hausdorff extension properties. Certain types of Hausdorff extension properties are shown to divide into classes such that each class has a largest member. In the latter part of the paper, for a Hausdorff extension property P satisfying one additionalproperty, the lattice of P-extensions of a fixed space X is related to X\X with a modified topology; this yields a theorem parallel to a similar result for the lattice of Hausdorff compactifications of a locally compact space X and βX\X obtained by Magill. © 1982, University of California, Berkeley. All Rights Reserved.