A tall space with a small bottom

István Juhász, Saharon Shelah, Lajos Soukup and Zoltán Szentmiklóssy

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 $ \kappa^{<\kappa} = \kappa$ then there is such a space of height $ \kappa^+$ with only $ \kappa$ many isolated points. This implies that there is a locally compact scattered space of height $ {\omega}_2$ with $ {\omega}_1$ isolated points in ZFC, solving an old problem of the first author.

Key words and phrases: locally compact scattered space, superatomic Boolean algebra

2000 Mathematics Subject Classification: 54A25, 06E05, 54G12, 03E20

Downloading the paper