## Piotr Niemiec

*Central subsets of Urysohn universal spaces*

Comment.Math.Univ.Carolinae 50,3 (2009) 445-461.**Abstract:**A subset $A$ of a metric space $(X,d)$ is central iff for every Kat\v{e}tov 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 Kat\v{e}tov maps of the real line are characterized.

**Keywords:** Urysohn's universal space, ultrahomogeneous spaces, extensions of isometries

**AMS Subject Classification:** 54E50 54D65

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