Embeddable properties of metric \(\sigma\)-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 \(\sigma\)-discrete spaces. Some related topics are also explo...

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Bibliographic Details
Published in:arXiv.org 2015-04
Main Authors: Plewik, Szymon, Walczyńska, Marta
Format: Article
Language:English
Subjects:
Chains
Subspaces
Online Access:Get full text
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Summary: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 \(\frak m\), 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.
ISSN:2331-8422