On uniformly discrete subsets in uniform spaces and topological groups

T. Banakh, S. Gabriyelyan, I. Protasov
Ivan Franko National University of Lviv, Ukraine; Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel; Department of Cybernetics, Kyiv University, Ukraine
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 third-named author for Polish groups. Applying the obtained results we extend the Hart-van Mill theorem ([9]) to all locally compact Abelian groups.
uniform space; uniformly locally compact uniform space; biuniform space; uniformly discrete subset; uniformly discrete number regular cardinal; bounded uniformly continuous function
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