Further consequences of
hard sets are explored in this paper, and some new relations between a space X and
its extension δX are shown. A generalization of perfect maps, called δ-perfect maps,
is introduced. It is found that among the WZ-maps, these are precisely the ones
which pull hard sets back to hard sels. Applications to δX are made. Maps which
carry hard sets to closed sets and maps which carry hard sets to hard sets are
considered, and it is seen that the image of a realcompact space under a
closed map is realcompact if and only if the map carries hard sets to hard
sets.
The last part of the paper introduces a generalization of normality, called
h-normal, in which disjoint hard sets are completely separated. It is found
that X is h-normal whenever vX is normal. The hereditary and productive
properties of h-normal spaces are investigated, and the h-normal spaces
are characterized in terms of δ-perfect WZ-maps. Finally as an analogue
of closed maps on normal spaces, a necessary and sufficient condition is
found that the image of an h-normal space under a δ-perfect WZ-map be
h-normal.
