The structures of Hausdorff metric in non-Archimedean spaces

Abstract

For non-Archimedean spaces $X$ and $Y,$ let ${\mathcal{M}}_{♭}(X),\mathfrak{M}(V\to W)$ and ${\mathfrak{D}}_{♭}(X,Y)$ be the ballean of $X$ (the family of the balls in $X$), the space of mappings from $X$ to $Y,$ and the space of mappings from the ballen of $X$ to $Y,$ respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., ${\hat{\rho }}_u,{\hat{\beta }}_{X,Y}^{\lambda },{\hat{\beta }}_{X,Y}^{\ast \lambda }$) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable $\lambda ,$ including some normed algebra structure. To some extent, the class ${\hat{\beta }}_{X,Y}^{\lambda }$ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when $X$ is compact and $Y=K$ is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of $K-$valued measures on $X.$

arXiv comment43 pages; this is the final version. Thanks to the anonymous referee's helpful comments, the original Theorem 2.10 is removed, Proposition 2.10 is stated now in a stronger form, the abstact is rewritten, the Monna-Springer is used in Section 5, and Theorem 5.2 is written in a more general form

References in corpus (1)

• Geometry of Non-Archimedean Gromov-Hausdorff distance

Cited by in corpus (1)

• Minimum spanning paths and Hausdorff distance in finite ultrametric spaces