The structures of Hausdorff metric in non-Archimedean spaces |
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Authors: | Derong Qiu |
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Institution: | 1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, P.R. China
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Abstract: | For non-Archimedean spaces X and Y, let $\mathcal{M}_\flat \left( X \right)$ , $\mathfrak{M}\left( {V \to W} \right)$ and $\mathfrak{D}_\flat \left( {X,Y} \right)$ 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 ballean 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., $\hat \rho _u$ , $\hat \beta _{X,Y}^\lambda$ , $\hat \beta _{X,Y}^{ * \lambda }$ ) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable λ, 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. |
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