On closed mappings of 𝜎-compact spaces and dimension

Abstract

A remainder of the Hilbert space l 2 l_{2} is a space homeomorphic to Z ∖ l 2 Z \setminus l_{2} , where Z Z is a metrizable compact extension of l 2 l_{2} , with l 2 l_{2} dense in Z Z . We prove that for any remainder K K of l 2 l_{2} , every non-one-point closed image of K K contains either a compact set with no transfinite dimension or compact sets of arbitrarily high inductive transfinite dimension i n d \mathrm {ind} . We shall also construct for each natural n n a σ \sigma -compact metrizable n n -dimensional space whose image under any non-constant closed map has dimension at least n n and analogous examples for the transfinite inductive dimension i n d \mathrm {ind} (this provides a rather strong negative solution of a problem in [Dissertationes Math. (Rozprawy Mat.) 216 (1983), pp. 1–41]).