Let be the set of all closed, convex and bounded subsets of a Banach space X equipped with the Hausdorff metric. In the first part of this work we study the density character of and investigate its connections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are concerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).
We give a new metrization theorem on terms of a new structure introduced by the authors in [2] and called fractal structure. As a Corollary we obtain Nagata-Smirnovs and Uryshons metrization Theorems.
We characterize those Tychonoff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family of nonempty compact subsets of . We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space is uniformly locally compact on if and only if is paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show...
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