Constructions of Lindelöf scattered P-spaces
Volume 259 / 2022
Abstract
We construct locally Lindelöf scattered P-spaces (LLSP spaces, for short) with prescribed widths and heights under different set-theoretic assumptions.
We prove that there is an LLSP space of width $\omega _1$ and height $\omega _2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $\omega _1$ and height $\omega _3$. Also, we prove a stepping up theorem which, for every cardinal $\lambda \geq \omega _2$, permits us to construct from an LLSP space of width $\omega _1$ and height $\lambda $ satisfying certain additional properties an LLSP space of width $\omega _1$ and height $\alpha $ for every ordinal $\alpha \lt \lambda ^+$. As consequences of the above results, we obtain the following theorems:
(1) For every ordinal $\alpha \lt \omega _3$ there is an LLSP space of width $\omega _1$ and height $\alpha $.
(2) It is relatively consistent with ZFC that there is an LLSP space of width $\omega _1$ and height $\alpha $ for every ordinal $\alpha \lt \omega _4$.
