Transfinite dimensions and mappings
First, an estimate is derived for the transfinite inductive dimensions Ind X and ind X of a normal space X, in terms of the dimension of a subset \(A\subset Y\) (for ind X assume A closed in X) and the relative dimension of its complement. Let \(d=Ind\) or ind. Then \[ dX\leq rd_ X(X- A)+dA+1\quad if\quad dA<\omega_ 0 \] and \[ dX\leq rd_ X(X-A)+dA\quad if\quad dA\geq \omega_ 0. \] (Recall that \(rd_ XY=\sup \{dF|\) F closed in X and \(F\subset Y\}\), for \(Y\subset X\) an arbitrary subset of X.) This result is then applied to prove that certain maps, e.g. closed maps of bounded multiplicity or open maps of countable multiplicity, preserve the property of a normal space that it has transfinite dimension, provided that the range of the map is hereditarily normal.
scientific article; zbMATH DE number 4065910