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

Spěvák, Jan

Commentat. Math. Univ. Carol. 56, No. 3, 377-399 (2015).

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)

Keywords:

covering dimension; topological group; function space; topology of pointwise convergence; free topological module; \(l\)-equivalence; \(G\)-equivalence

Citations:

Zbl 0518.54030