Characterizations of Urysohn universal ultrametric spaces

Abstract In this article, using the existence of infinite equidistant subsets of closed balls, we first characterize the injectivity of ultrametric spaces for finite ultrametric spaces. This method also gives characterizations of the Urysohn universal ultrametric spaces. As an application, we find that the operations of the Cartesian product and the hyperspaces preserve the structures of the Urysohn universal ultrametric spaces. Namely, let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,d) be the Urysohn universal ultrametric space. Then, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>×</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>×</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X\times X,d\times d) is isometric to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,d) , and show that the hyperspace consisting of all non-empty compact subsets of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,d) and symmetric products of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,d) are isometric to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,d) . We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,d) .