Metric Topologies on Multiset Spaces as Topological Monoids and Their Group Completion
Donghan Kim
TL;DR
The paper develops a metrizable multiset framework for metric spaces by introducing a matching distance $d_{\mathbb{N}[X]}$ on the infinite symmetric product $\SP(X)$, yielding the metrizable abelian topological monoid $\mathbb{N}[X]$ that embeds $X$ isometrically. It extends to the free abelian group $\mathbb{Z}[X]$ with a compatible metric, making it a metrizable abelian topological group and preserving isometric embeddings from $X$ through $\mathbb{N}[X]$ to $\mathbb{Z}[X]$. The authors identify the completion of $\mathbb{N}[X]$ as $\overline{\mathbb{N}}[X]$ with the extended distance $d_\ell$, showing $\mathbb{N}[X]$ is dense and completing under mild conditions on $X$. A quotient-metric construction is also developed, ensuring compatibility between quotient topology and metric structure, which provides a flexible toolkit for analyzing metric multisets, group completions, and related topological-algebraic structures.
Abstract
We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, 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 $\mathbb{Z}[X]$, where a metric $d_{\mathbb{Z}[X]}$ induces a metrizable abelian topological group structure. We further identify the metric completion of $\mathbb{N}[X]$, showing that it carries a canonical extension of the matching metric.