Module-valued functors preserving the covering dimension

Commentationes mathematicae Universitatis Carolinae

Jan Spěvák¹ 

关键词: covering dimension;    topological group;    function space;    topology of pointwise convergence;    free topological module;    $l$-equivalence;    $G$-equivalence;   

DOI  :  10.14712/1213-7243.2015.131

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

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 [{\it The coincidence of the dimensions dim of $l$-equivalent spaces\/}, Soviet Math. Dokl. {\bf 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 $G$.

【 授权许可】

CC BY