Scattered $P$-spaces of weight $\omega_1$

TL;DR

This paper explores the properties and embeddings of scattered P-spaces of weight ω₁, revealing similarities with separable metric spaces and establishing their embedding into ω₂.

Contribution

It introduces the study of dimensional types of scattered P-spaces of weight ω₁ and shows their embedding into ω₂, highlighting parallels with separable metric spaces.

Findings

Scattered P-spaces of weight ω₁ can be embedded into ω₂.

Dimensional types of these P-spaces resemble those of separable metric spaces.

Established similarities for spaces with Cantor–Bendixson rank less than ω₁.

Abstract

We examine dimensional types of scattered $P$-spaces of weight $\omega_1$. Such spaces can be embedded into $\omega_2$. There are established similarities between dimensional types of scattered separable metric spaces and dimensional types of $P$-spaces of weight $\omega_1$ with Cantor--Bendixson rank less than $\omega_1$.