Central subsets of Urysohn universal spaces

Piotr Niemiec

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 3, page 445-461
  • ISSN: 0010-2628

Abstract

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A subset A of a metric space ( X , d ) is central iff for every Katětov map f : X → ℝ upper bounded by the diameter of X and any finite subset B of X there is x ∈ X such that f ( a ) = d ( x , a ) for each a ∈ A ∪ B . Central subsets of the Urysohn universal space 𝕌 (see introduction) are studied. It is proved that a metric space X is isometrically embeddable into 𝕌 as a central set iff X has the collinearity property. The Katětov maps of the real line are characterized.

How to cite

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Niemiec, Piotr. "Central subsets of Urysohn universal spaces." Commentationes Mathematicae Universitatis Carolinae 50.3 (2009): 445-461. <http://eudml.org/doc/33327>.

@article{Niemiec2009,
abstract = {A subset $A$ of a metric space $(X,d)$ is central iff for every Katětov map $f: X \rightarrow \mathbb \{R\}$ 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.},
author = {Niemiec, Piotr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Urysohn's universal space; ultrahomogeneous spaces; extensions of isometries; Urysohn's universal space; ultrahomogeneous space; extension; of isometries},
language = {eng},
number = {3},
pages = {445-461},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Central subsets of Urysohn universal spaces},
url = {http://eudml.org/doc/33327},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Niemiec, Piotr
TI - Central subsets of Urysohn universal spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 3
SP - 445
EP - 461
AB - A subset $A$ of a metric space $(X,d)$ is central iff for every Katětov map $f: X \rightarrow \mathbb {R}$ 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.
LA - eng
KW - Urysohn's universal space; ultrahomogeneous spaces; extensions of isometries; Urysohn's universal space; ultrahomogeneous space; extension; of isometries
UR - http://eudml.org/doc/33327
ER -

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