Vol. 79, No. 2, 1978

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Maps and h-normal spaces

Marlon C. Rayburn

Vol. 79 (1978), No. 2, 549–561

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.

Mathematical Subject Classification 2000
Primary: 54D40
Received: 5 December 1977
Revised: 11 April 1978
Published: 1 December 1978
Marlon C. Rayburn