A topological property 𝒫 is
called a Hausdorff extension property if each Hausdorff space X can be densely
embedded in a space κ𝒫X such that (1) κ𝒫X is a Hausdorff space with
property 𝒫, (2) if Y is a Hausdorff extension of X with 𝒫, then there is a
continuous function f = κ𝒫X → Y such that f(x) = x for each x ∈ X, and (3) if
X is H-closed, then κ𝒫X = 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 additional property, the lattice of
𝒫-extensions of a fixed space X is related to κ𝒫X ∖ 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.
