## 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

### Citations:

Zbl 0999.54022; Zbl 0468.57015
Full Text:

### References:

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