On uniformly discrete subsets in uniform spaces and topological groups 

Author 
t.o.banakh@gmail.com, saak@math.bgu.ac.il, i.v.protasov@gmail.com
Ivan Franko National University of Lviv, Ukraine; Department of Mathematics,
BenGurion University of the Negev, BeerSheva, Israel; Department of Cybernetics, Kyiv University, Ukraine

Abstract 
We study uniform spaces making use of their uniformly discrete subsets. For a uniform space $(X,\mathcal{U})$, we define the uniformly discrete number $\mathop{\rm ud}(X)$ of $X$ as the supremum of cardinalities of its uniformly discrete subsets. It is shown that $\mathop{\rm ud}(X)$ coincides with the index of narrowness $\mathop{\rm ib}(X)$ of $X$. Using uniformly discrete subsets of uniform spaces we characterize regular cardinals. For a complete metrizable biuniform space $(X,\mathcal{L},\mathcal{R})$ it is proved the equivalence of the {following} assertions: (a) the spaces of all bounded uniformly continuous real valued functions on $(X, \mathcal{L})$ and $(X, \mathcal{R})$ coincide; (b) $(X, \mathcal{L})$ and $(X, \mathcal{R})$ have the same families of uniformly discrete subsets; (c) $\mathcal{L} = \mathcal{R}$. This result generalizes the result obtained by the thirdnamed author for Polish groups. Applying the obtained results we extend the Hartvan Mill theorem ([9]) to all locally compact Abelian groups.

Keywords 
uniform space; uniformly locally compact uniform space; biuniform space; uniformly discrete
subset; uniformly discrete number regular cardinal; bounded uniformly continuous function

DOI 
doi:10.15330/ms.45.1.7697

Reference 
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Pages 
7697

Volume 
45

Issue 
1

Year 
2016

Journal 
Matematychni Studii

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