On certain equivalences of metric spaces
Authors
Year
2025
Published
Proceedings of the American Mathematical Society. 2025, 153(1), 239-249. ISSN 0002-9939.
Type
Article
Departments
Annotation
The normed spaces of molecules constructed by Arens and Eells allow us to define two natural equivalence relations on the class of complete metric spaces. We say that two complete metric spaces M and N are M at- equivalent if their normed spaces of molecules are isomorphic and we say that they are .F- equivalent if the corresponding completions - the Lipschitz-free Banach spaces .F ( M ) and .F ( N ) - are isomorphic. In this note, we compare these and some other relevant equivalences of metric spaces. Clearly, M at-equivalent spaces are .F-equivalent. Our main result states that M at-equivalent spaces must have the same covering dimension. In combination with the work of Godard, this implies that M at-equivalence is indeed strictly stronger than .F-equivalence. However, M at-equivalent spaces need not be homeomorphic, as we demonstrate through a general construction. We also observe that M at-equivalence does not preserve the Assouad dimension. We introduce a natural notion of a free basis to simplify the notation.
Topological groups with invariant linear spans
Authors
Year
2022
Published
Revista Matemática Complutense. 2022, 35(1), 219-226. ISSN 1139-1138.
Type
Article
Departments
Annotation
Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let Z(A) be the direct sum of |A|-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group Z(A) has invariant linear span. This answers a question from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. We prove that given a non-discrete sequential space X, the free abelian topological group A(X) over X is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.
Topologically independent sets in precompact groups
A note on multiplier convergent series
Direct sums and products in topological groups and vector spaces
Authors
Dikranjan, D.; Shakhmatov, D.; Spěvák, J.
Year
2016
Published
Journal of Mathematical Analysis and Applications. 2016, 1257-1282. ISSN 0022-247X.
Module-valued functors preserving the covering dimension
Authors
Year
2015
Published
Commentationes Mathematicae Universitatis Carolinae. 2015, ISSN 0010-2628.
Productivity of sequences in non-abelian topological groups
Authors
Year
2015
Published
Topology and Its Applications. 2015, 163-177. ISSN 0166-8641.
Type
Article
Departments
FINITE-VALUED MAPPINGS PRESERVING DIMENSION
Type
Article
Departments
Productivity of sequences with respect to a given weight function
Authors
Spěvák, J.; Shakhmatov, D.; Dikranjan, D.
Year
2011
Published
Topology and Its Applications. 2011, 2011(158), ISSN 0166-8641.
Group-valued continuous functions with the topology of pointwise convergence
Authors
Spěvák, J.; Shakhmatov, D.
Year
2010
Published
Topology and Its Applications. 2010, 2010(157), 1518-1540. ISSN 0166-8641.