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The Urysohn universal metric space is homeomorphic to a Hilbert space. (English) Zbl 1062.54036

Call a separable metric space \(X\) {universal} if it contains an isometric copy of every separable metric space, and {ultrahomogeneous} if every isometry between two finite metric subspaces of \(X\) can be extended to an isometry of \(X\) onto itself. The Urysohn space \(U\) is the unique (up to isometry) universal ultrahomogeneous space.
In the first part of this paper some equivalent definitions of \(U\), as well as recent results concerning this space, are briefly surveyed. In the second part the author proves that \(U\) is homeomorphic to the Hilbert space \(\ell^2\) (and hence, to any separable infinite-dimensional Fréchet space).
The proof is based on the fact that \(U\), as shown by S. A. Bogatyi [Russ. Math. Surv. 55, No. 2, 332–333 (2000; Zbl 0999.54022); translation from Usp. Mat. Nauk 55, No. 2, 131–132 (2000)] is compactly injective, that is, every continuous map from a compact metric space \(K\) to \(U\) can be extended over any compact metric superspace of \(K\); and on H. Toruńczyk’s characterization of \(\ell^2\) as the unique (up to homeomorphisms) separable metrizable AR space with the discrete approximation property [Fundam. Math. 111, 247–262 (1981; Zbl 0468.57015)].

MSC:

54F65 Topological characterizations of particular spaces
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54E50 Complete metric spaces
54E35 Metric spaces, metrizability
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