Abstract
The equality of the dimensions \(\operatorname{Ind}X\) and \(\operatorname{dim}X\) of a first countable paracompact \(\sigma\)-space \(X\) with a 1-continuous semimetric is proved. A partial positive answer to A. V. Arkhangel’skii’s question about the equality of dimensions for first countable spaces with a countable network is given. As a consequence, the equality of the dimensions \(\operatorname{Ind}X\) and \(\operatorname{dim}X\) for Nagata spaces (that is, stratifiable first countable spaces) with a 1-continuous semimetric is obtained.
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Acknowledgments
It is a pleasure to thank Professor A. V. Arkhangel’skii for attention and useful discussions.
Funding
This work was supported by the Russian Science Foundation under grant 19-11-00223.
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 499–516 https://doi.org/10.4213/mzm13669.
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Leibo, I.M. Equality of Dimensions for Some Paracompact \(\sigma\)-Spaces. Math Notes 113, 488–501 (2023). https://doi.org/10.1134/S0001434623030215
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DOI: https://doi.org/10.1134/S0001434623030215