Embeddable properties of metric $\sigma$-discrete spaces

Abstract

Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpiński 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 $\mathfrak{m}$, there exist $2^{\mathfrak{m}}$ many non-homeomorphic metric scattered spaces of the cardinality $\mathfrak{m}$; 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.

20 pages