Introduction. A Hausdorff
topological space X is called ultra-Hausdorff if, given two distinct points p and q of
X, there is an open-and-closed (henceforth called “clopen”) subset A of X
such that p ∈ A and q∉A. A Hausdorff space X is H-closed if, whenever
it is embedded as a subspace of another Hausdorff space Y , it is a closed
subset of Y . In this paper we characterize those Hausdorff spaces that have
ultra-Hausdorff H-closed extensions and construct, for such spaces, the projective
maximum of the set of ultra-Hausdorff H-closed extensions. We compare
this projective maximum to the Katětov H-closed extension, and examine
when continuous functions can be continuously extended to this projective
maximum.
