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Note on isometric universality and dimension. (English) Zbl 1329.54033
The authors solve some problems raised by S. D. Iliadis in [Topology Appl. 160, No. 11, 1271–1283 (2013; Zbl 1282.54013); in: Proceedings of the international conference on topology and its applications (ICTA 2011), Islamabad, Pakistan, July 4–10, 2011. Cambridge: Cambridge Scientific Publishers. 231–241 (2012; Zbl 1300.54004)] by proving the following two theorems: 1) If a separable complete metric space \(M\) contains isometric copies of all countable complete metric spaces then it contains isometric copies of all separable metric spaces. 2) If a separable complete metric space contains a uniformly homeomorphic copy of every countable topologically discrete complete metric space then it contains a uniformly homeomorphic copy of every separable metric space.
The proofs make clever use of the fact that the hyperspace of compact subsets of the space of rational numbers (with the Hausdorff metric) is not analytic. This is used to build a separable complete metric space \(E\) with the property that every analytic subset of its hyperspace that contains all countable closed sets must also contain an isometrically universal metric space, and again in the hyperspace of the product of \(E\times M\) to find the graph of an isometry of such a universal space into \(M\).
Reviewer: K. P. Hart (Delft)

54E50 Complete metric spaces
54B20 Hyperspaces in general topology
54F45 Dimension theory in general topology
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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