Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function U → U is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the defin...

We show that a finite metric space A admits an extension to a finite metric space B so that each partial isometry of A extends to an isometry of B. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space.

Prompted by a recent question of G. Hjorth [12] as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we answer this question and more generally investigate partitions of countable metric spaces. We show that an indivisible metric space must be totally Cantor...

This paper shows a practical way of modelling non-linear dynamic systems with one input using the discrete form of the integral operator of the Urysohn type. A new iterative procedure for the identification of the Uryshon operator is proposed. It represents a sequence of elementary computational steps providing iterative improvements for the Urysohn model using observable input-output data. The...

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (Q,+) by measure preservi...

Keywords: Functional integral equation Existence Monotonic solutions Urysohn Compact in measure a b s t r a c t We study the existence of monotonic solutions for a perturbed functional integral equation of Urysohn type in the space of Lebesgue integrable functions on an unbounded interval. The technique associated with measures of noncompactness (in both the weak and the strong sense) and the D...

We call a nonempty subset A of a topological space X finitely non-Urysohn if for every nonempty finite subset F of A and every family {Ux : x ∈ F} of open neighborhoods Ux of x ∈ F , ∩{cl(Ux) : x ∈ F} 6 = ∅ and we define the non-Urysohn number of X as follows: nu(X) := 1 + sup{|A| : A is a finitely non-Urysohn subset of X}. Then for any topological space X and any subset A of X we prove the fol...

philosophers from one hand take the “al-vahed” axiom for granted, and on the other hand, they assume that the almighty to be of absolute power. now the question is whether: this is at odds with the “al- vahed” axiom? how can we accept at the same time the absolute power of the almighty and the “al-vahed” axiom? we should say here: majority of philosophers, mystics and kalam specialists have cla...