Bibliographic Information: Leibo, I. M. (2024). Coincidence of the Dimensions of First Countable Spaces with a Countable Network. arXiv preprint arXiv:2410.03469v1.
Research Objective: This paper aims to address a long-standing question posed by Arkhangel'skii: whether the three classical topological dimension functions—ind, Ind, and dim—coincide for first countable spaces possessing a countable network.
Methodology: The author employs concepts from dimension theory, particularly focusing on the properties of first countable paracompact σ-spaces. The proof leverages the equivalence of certain conditions related to dimension in metrizable spaces and extends them to the broader class of first countable paracompact σ-spaces. A crucial aspect of the proof involves demonstrating the existence of an everywhere f-system within these spaces, drawing upon the concept of almost semicanonical spaces.
Key Findings: The paper's central result is the proof that for any first countable paracompact σ-space X, the dimensions dim X and Ind X are equal. This finding is further strengthened by establishing the equivalence of four conditions related to the dimensions of such spaces.
Main Conclusions: The study provides a positive answer to Arkhangel'skii's question by proving the equality of ind, dim, and Ind dimensions for first countable spaces with a countable network. This conclusion stems from the more general result concerning the coincidence of dim and Ind dimensions in first countable paracompact σ-spaces.
Significance: This paper significantly contributes to dimension theory within topology by resolving a question that has remained open for over 50 years. It deepens the understanding of dimensional relationships in topological spaces with specific properties.
Limitations and Future Research: While the paper successfully addresses Arkhangel'skii's question for first countable spaces with a countable network, the broader question of whether the dimensions coincide for all topological groups with a countable network remains open. The author suggests investigating the properties of almost semicanonical spaces, particularly in the context of topological groups, as a potential avenue for future research to address this open question.
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by I.M. Leibo at arxiv.org 10-07-2024
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