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 ${\mathbf {N}}$-compact, complete metrizable spaces with $\operatorname {ind} = 0$ and $\operatorname {Ind} = 1$ are provided, answering questions of Mrowka and Roy. An example with weight $\mathfrak {c}$ and positive Ind such that subspaces with smaller weight have $\operatorname {Ind} = 0$ is produced in ZFC. Assuming an additional axiom, for each cardinal $\lambda$ a space of positive Ind with all subspaces with weight less than $\lambda$ strongly zero-dimensional is constructed.