Constructions of Lindelöf scattered P-spaces

Abstract

We construct locally Lindelöf scattered P-spaces (LLSP spaces, in 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 that, for every cardinal $\lambda \ge {\omega }_2$, 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 $\alpha <{\lambda }^{+}$. Then, we obtain as consequences of the above results the following theorems: (1) For every ordinal $\alpha <{\omega }_3$ 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 $\alpha <{\omega }_4$.

14 pages