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.
