Vol. 103, No. 1, 1982

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Extensions of Hausdorff spaces

Jack Ray Porter and R. Grant Woods

Vol. 103 (1982), No. 1, 111–134

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.

Mathematical Subject Classification 2000
Primary: 54C20
Secondary: 54D25
Received: 29 May 1981
Published: 1 November 1982
Jack Ray Porter
R. Grant Woods