On uniformly discrete subsets in uniform spaces and topological groups

We study uniform spaces making use of their uniformly discrete subsets. For a uniform space (X,U), we define the uniformly discrete number ud(X) of X as the supremum of cardinalities of its uniformly discrete subsets. It is shown that ud(X) coincides with the index of narrowness ib(X) of X. Using uniformly discrete subsets of uniform spaces we characterize regular cardinals. For a complete metrizable biuniform space (X,L,R) it is proved the equivalence of the {following} assertions: (a) the spaces of all bounded uniformly continuous real valued functions on (X,L) and (X,R) coincide; (b) (X,L) and (X,R) have the same families of uniformly discrete subsets; (c) L=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.