D-forced spaces: a new approach to resolvability

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 &#34;full control&#34; over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal $λ$ a 0-dimensional $T_2$, hence Tychonov, space which is $μ$-resolvable for all $μ< λ$ but not $λ$-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are $ω$-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning resolvability as well.