Embeddable properties of metric $\sigma$-discrete spaces

TL;DR

This paper explores the properties and classifications of metric scattered spaces, extending known theorems and analyzing embeddability and dimensional types in both countable and uncountable contexts.

Contribution

It generalizes embeddable properties from countable to uncountable metric iscrete spaces and revises key theorems in the field.

Findings

Existence of 2^{} non-homeomorphic metric scattered spaces of cardinality 

Poset of dimensional types on stationary subsets of  contains uncountable chains and anti-chains

Revised proofs of Mazurkiewicz-Sierpi
42ski and Knaster-Urbanik theorems

Abstract

Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric $\sigma$-discrete spaces. Some related topics are also explored. For example: For each infinite cardinal number $\frak m$, there exist $2^{\frak m}$ many non-homeomorphic metric scattered spaces of the cardinality $\frak m $; If $X \subseteq \omega_1$ is a stationary set, then the poset formed from dimensional types of subspaces of $X$ contains uncountable anti-chains and uncountable strictly descending chains.