MAPPINGS AND REALCOMPACT SPACES

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 mapsmay 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 Ø is a closed map from X onto a weak cb, q-space Y, then X is realcompact implies that Y is realcompact. © 1969 by Pacific Journal of Mathematics.