Vol. 33, No. 3, 1970

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Generalizations of realcompact spaces

Nancy Dykes

Vol. 33 (1970), No. 3, 571–581

Two generalizations of realcompactness are examined; a-realcompactness and c-realcompactness. The first, a-realcompactness, is invariant under perfect maps and is a generalization of almost realcompactness. Spaces that are a-realcompact and cb, almost realcompact and weak cb, or c-realcompact and weak cb, are also realcompact. Using these properties we obtain the following three theorems. If X is weak cb, then the union of two closed realcompact spaces is realcompact. The union of a countable collection of open sets is realcompact, if the closure of each open set is realcompact. If X is cb, the union of a countable collection of realcompact spaces is realcompact. The latter statement has been shown for X normal and the subspaces closed by Mrowka. It is not known if normality implies weak cb. The problem of preserving realcompactness under closed maps is also considered. Using a-realcompactness, we obtain the following special case. Realcompactness is preserved under closed maps if the range is a cb, k-space.

Mathematical Subject Classification
Primary: 54.52
Received: 11 June 1969
Published: 1 June 1970
Nancy Dykes