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

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.