The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces $ X $ and $ Y, $ let $ \mathcal{M}_{\flat } (X), \mathfrak{M}(V \rightarrow W) $ and $ \mathfrak{D}_{\flat }(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., $ \widehat{\rho } _{u}, \widehat{\beta }_{X, Y}^{\lambda }, \widehat{\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 $ \widehat{\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. $