A tall space with a small bottom

Abstract

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if κ > κ = κ \kappa ^{>\kappa } = \kappa , then there is such a space of height κ + \kappa ^+ with only κ \kappa many isolated points. This implies that there is a locally compact scattered space of height ω 2 {\omega }_2 with ω 1 \omega _1 isolated points in ZFC, solving an old problem of the first author.