Closed maps and spaces with zero-dimensional remainders (Q1069489)

A 0-space is a completely regular Hausdorff space possessing a compactification with zero-dimensional remainder. It is well-known that any 0-space X possesses a maximum compactification \(F_ 0X\) having this property. The following question is considered: if \(f: X\to Y\) is a closed map, and X, Y are 0-spaces, under what conditions on X, Y and/or f will f extend to \(g\in C\) \((F_ 0X,F_ 0Y)?\) It is proved that if Y is rimcompact, then it is (necessary and) sufficient that for any distinct pair of points y,z\(\in Y\), \(Cl_{F_ 0X}f^{-1}(y)\cap Cl_{F_ 0X}f^{-1}(z)=\emptyset\). This result is used to show that if i) X is a realcompact or metacompact 0-space and Y is a rimcompact space in which the set of q-points has discrete complement, or if ii) X is a metacompact 0-space or a locally compact realcompact space, and Y is a rimcompact k- space, then any closed map from X into Y extends to a map from \(F_ 0X\) into \(F_ 0Y\).