## Module-valued functors preserving the covering dimension.(English)Zbl 1349.54070

Summary: We prove a general theorem about preservation of the covering dimension $$\dim$$ by certain covariant functors that implies, among others, the following concrete results.
(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 $$\dim X=\dim Y$$.
(ii) If free precompact abelian groups of Tychonoff spaces $$X$$ and $$Y$$ are topologically isomorphic, then $$\dim X=\dim Y$$.
(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 $$\dim X=\dim Y$$.
The classical result of V. G. Pestov [Sov. Math., Dokl. 26, 380–383 (1982; Zbl 0518.54030)] about preservation of the covering dimension by $$l$$-equivalence immediately follows from item (i) by taking the topological group of real numbers as $$G$$.

### MSC:

 54F45 Dimension theory in general topology 54H11 Topological groups (topological aspects) 54H13 Topological fields, rings, etc. (topological aspects)

Zbl 0518.54030
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