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Embeddable properties of metric $σ$-discrete spaces

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 $σ$-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 $X \subseteq ω_1$ is a stationary set, then the poset formed from dimensional types of subspaces of $X$ contains uncountable anti-chains and uncountable strictly descending chains.