Prethick subsets and partitions of metric spaces

Author K. D. Protasova
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
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Pages 115-117
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
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