The problem of preserving
realcompactness under perfect and closed maps is studied. The main result is that
realcompactness is preserved under closed maps if the range is a normal, weak cb,
k-space. Generalizing a result of Frollk, we show that realcompactness is preserved
under perfect maps if the range is weak cb. Moreover, the problem of preserving
realcompactness under perfect maps may be reduced to the following: When does the
absolute of X being realcompact imply that X is realcompact? Likewise the problem
of preserving topological completeness under perfect maps may be reduced to an
analogous question. The following special case is also proved. If ϕ is a closed map
from X onto a weak cb, q-space Y , then X is realcompact implies that Yis
realcompact.
