Module-valued functors preserving the covering dimension

Spěvák, Jan

Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 3, pp. 377-399.

Résumé

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 $G$.

Détail

DOI : 10.14712/1213-7243.2015.131

Classification : 54H11, 54H13
Keywords: covering dimension; topological group; function space; topology of pointwise convergence; free topological module; $l$-equivalence; $G$-equivalence