Geodesic
Central subsets of Urysohn universal spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 445-461.

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A subset $A$ of a metric space $(X,d)$ is central iff for every Katětov map $f: X \to \mathbb R$ upper bounded by the diameter of $X$ and any finite subset $B$ of $X$ there is $x\in X$ such that $f(a) = d(x,a)$ for each $a\in A \cup B$. Central subsets of the Urysohn universal space $\mathbb U$ (see introduction) are studied. It is proved that a metric space $X$ is isometrically embeddable into $\mathbb U$ as a central set iff $X$ has the collinearity property. The Katětov maps of the real line are characterized.
Classification : 54D65, 54E50
Keywords: Urysohn's universal space; ultrahomogeneous spaces; extensions of isometries 
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     title = {Central subsets of {Urysohn} universal spaces},
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Niemiec, Piotr. Central subsets of Urysohn universal spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 445-461. http://geodesic.mathdoc.fr/item/CMUC_2009__50_3_a10/