D-forced spaces: a new approach to resolvability

TL;DR

This paper introduces a ZFC-based method to construct special dense subspaces of Cantor cubes, allowing precise control over dense subsets and solving longstanding problems about the limits of resolvability in topological spaces.

Contribution

It presents a novel ZFC method for constructing spaces with specific resolvability properties, including the first solution to a 1967 problem about maximal resolvability.

Findings

Constructed spaces that are $orall \, ext{uncountable regular } \\lambda$, $ ext{not } \\lambda$-resolvable.

Resolved the 1967 question on whether $\\omega$-resolvable spaces are maximally resolvable.

Enabled solutions to several open problems in resolvability theory.

Abstract

We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal $\lambda$ a 0-dimensional $T_2$, hence Tychonov, space which is $\mu$-resolvable for all $\mu < \lambda$ but not $\lambda$-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are $\omega$-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning resolvability as well.