Constructions of Lindel\"{o}f scattered P-spaces

TL;DR

This paper constructs locally Lindel"of scattered P-spaces with specific widths and heights, demonstrating their existence under various set-theoretic assumptions and establishing consistency results within ZFC.

Contribution

It introduces new constructions of LLSP spaces with prescribed widths and heights, including consistency results and a stepping-up theorem for building larger spaces.

Findings

Existence of LLSP space with width ω₁ and height ω₂.

Consistency of LLSP space with width ω₁ and height ω₃.

Construction of LLSP spaces with height less than ω₄ for all ordinals below ω₄.

Abstract

We construct locally Lindel\"of 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 $\lambda \geq \omega_2$, 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 $\alpha < \lambda^+$. 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 $\alpha < \omega_4$.