We investigate the space of convex minimal usco maps from a Tychonoff space to the space of real numbers. Its elements are set-valued maps that are important e.g. in the study of subdifferentials of convex functions. We show that if the underlying space is normal, convex minimal usco maps can be approximated in the Vietoris topology by continuous functions. Using the strong Choquet game we prove complete metrizability of the space of convex minimal usco maps equipped with the upper Vietoris topology. We also study first countability, second countability and other properties of the (upper) Vietoris topology on this space.

中文翻译：

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凸最小Usco映射空间的拓扑性质。

我们研究了从Tychonoff空间到实数空间的凸极小usco映射的空间。它的元素是集值映射，在例如凸函数的微分研究中很重要。我们表明，如果基础空间是正常的，则凸维最小usco映射可以通过连续函数在Vietoris拓扑中近似。使用强大的Choquet游戏，我们证明了配有上Vietoris拓扑的凸极小usco映射空间的完全可度量性。我们还研究了该空间上（上部）维耶托里斯拓扑的第一可数性，第二可数性和其他属性。