# Prethick subsets and partitions of metric spaces

Author K. D. Protasova
islab@unicyb.kiev.ua
Department of Cybernetics, Kyiv University

Abstract A subset $A$ of a metric space $(X,d)$ is called thick if, for every $r>0$, there is $a\in A$ such that $B_{d}(a,r)\subseteq A,$ where $B_{d}(a,r)=\{x\in X\colon d(x,a)\leq r\}$. We show that if $(X, d)$ is unbounded and has no asymptotically isolated balls then, for each $r>0$, there exists a partition $X=X_{1}\cup X_{2}$ such that $B_{d}(X_{1},r)$ and $B_{d}(X_{2},r)$ are not thick.
Keywords metric space; thick and prethick subsets; asymptotically isolated balls
Reference 1. T. Banakh, I.V. Protasov, S. Slobodianiuk, Subamenable groups and their partitions, preprint (http:// arxiv.org/abs/1210.5804).

2. T. Banakh, I. Zarichnyi, The coarse characterization of homogeneous ultrametric space, Groups, Geometry and Dynamics, 5 (2011), 691–728.

3. I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math. Stud. Monogr. Ser., V.11, VNTL Publisher, Lviv, 2003.

4. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser. V.12, VNTL Publisher, Lviv, 2007.

Pages 115-117
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
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