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