Metric Topologies on Multiset Spaces as Topological Monoids and Their Group Completion

2025/10/11 by Kim, Donghan
#FOS: Mathematics #Metric Geometry (math.MG) #Primary 54E35 #Secondary 54H11

paper · doi:10.48550/arxiv.2510.10080

Abstract

We construct a multiset space ℕ[X] over a metric space X that simultaneously enjoys desirable topological properties and admits a natural matching metric d_ℕ[X], making it a metrizable abelian topological monoid whose structure is compatible with the original metric on X. This framework extends naturally to the free abelian group ℤ[X], where a metric d_ℤ[X] induces a metrizable abelian topological group structure. We further identify the metric completion of ℕ[X], showing that it carries a canonical extension of the matching metric.

Citations

• Partial monoids and Dold-Thom functors

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