Metrizable spaces where the inductive dimensions disagree

Abstract

A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roy’s technique, these spaces can often be shown to have positive large inductive dimension. Examples of N {\mathbf {N}} -compact, complete metrizable spaces with ind = 0 \operatorname {ind} = 0 and Ind = 1 \operatorname {Ind} = 1 are provided, answering questions of Mrowka and Roy. An example with weight c \mathfrak {c} and positive Ind such that subspaces with smaller weight have Ind = 0 \operatorname {Ind} = 0 is produced in ZFC. Assuming an additional axiom, for each cardinal λ \lambda a space of positive Ind with all subspaces with weight less than λ \lambda strongly zero-dimensional is constructed.