On closed mappings of 𝜎-compact spaces and dimension

Abstract

A remainder of the Hilbert space l 2 is a space homeomorphic to Z \l 2 , where Z is a metrizable compact extension of l 2 , with l 2 dense in Z. We prove that for any remainder K of l 2 , every non-one-point closed image of K either contains a compact set with no transfinite dimension or contains compact sets of arbitrarily high inductive transfinite dimension ind. We shall also construct for each natural n a σ-compact metrizable n-dimensional space whose image under any non-constant closed map has dimension at least n, and analogous examples for the transfinite inductive dimension ind (this provides a rather strong negative solution of a problem in [EnP]).