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Title: Module-valued functors preserving the covering dimension (English)
Author: Spěvák, Jan
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 56
Issue: 3
Year: 2015
Pages: 377-399
Summary lang: English
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Category: math
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Summary: We prove a general theorem about preservation of the covering dimension $\operatorname{dim}$ by certain covariant functors that implies, among others, the following concrete results. \begin{enumerate} \item[(i)] If $G$ is a pathwise connected separable metric NSS abelian group and $X$, $Y$ are Tychonoff spaces such that the group-valued function spaces $C_p(X,G)$ and $C_p(Y,G)$ are topologically isomorphic as topological groups, then $\operatorname{dim} X=\operatorname{dim} Y$. \item[(ii)] If free precompact abelian groups of Tychonoff spaces $X$ and $Y$ are topologically isomorphic, then $\operatorname{dim} X=\operatorname{dim} Y$. \item[(iii)] If $R$ is a topological ring with a countable network and the free topological $R$-modules of Tychonoff spaces $X$ and $Y$ are topologically isomorphic, then $\operatorname{dim} X=\operatorname{dim} Y$. \end{enumerate} The classical result of Pestov [The coincidence of the dimensions dim of $l$-equivalent spaces, Soviet Math. Dokl. 26 (1982), no. 2, 380--383] about preservation of the covering dimension by $l$-equivalence immediately follows from item (i) by taking the topological group of real numbers as $G$. (English)
Keyword: covering dimension
Keyword: topological group
Keyword: function space
Keyword: topology of pointwise convergence
Keyword: free topological module
Keyword: $l$-equivalence
Keyword: $G$-equivalence
MSC: 54H11
MSC: 54H13
idZBL: Zbl 06487001
idMR: MR3390284
DOI: 10.14712/1213-7243.2015.131
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Date available: 2015-07-09T20:54:54Z
Last updated: 2017-10-02
Stable URL: http://hdl.handle.net/10338.dmlcz/144352
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