On uniformly discrete subsets in uniform spaces and
topological groups

T. Banakh; S. Gabriyelyan; I. Protasov

doi:10.15330/ms.45.1.76-97

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.

1. A.V. Arhangelskii, M.G. Tkachenko, Topological groups and related strutures, Atlantis Press/World Scientific, Amsterdam-Raris, 2008.

2. W.W. Comfort, S. Hernandez, F.J. Trigos-Arrieta, Relating a locally compact Abelian group to its Bohr compactification, Adv. Math., 120 (1996), 322-344.

3. E.K. van Douwen, The maximal totally bounded group topology an G and the biggest minimal G-space, for abelian groups G, Topology Appl., 34 (1990), 69-91.

4. V. Efremovich, The geometry of proximity, Math. Sb., 73 (1952), 189-200.

5. R. Engelking, General Topology, PWN, Polish Scientific Publ, Warszawa, 1977.

6. L. Gillman, M. Jerison, Rings of continuous functions, Van Nostrand, New York, 1960.

7. I.I. Guran, On topological groups close to being LindelЃNof, Soviet. Math. Dokl., 23 (1981), 173-175.

8. I.I. Guran, On embeddings of topological groups, available in VINITI, Moskow, №1483-81, 1981.

9. K.P. Hart, J. van Mill, Discrete sets and the maximal totally bounded group topology, J. Pure Appl. Algebra, 70 (1991), 73-80.

10. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, V.I, 2nd ed. Springer-Verlag, Berlin, 1979.

11. G. Itzkowitz, Functional balance, discrete balance, and balance in topological groups, Topology Proc., 28 (2004), 569-577.

12. M. Katetov, On real valued functions in topological spaces, Fund. Math., 38 (1951), 85-91.

13. M. Megrelishvili (Levy), P. Nickolas, V. Pestov, Uniformities and uniformly continuous functions on locally connected groups, Bull. Austral. Math. Soc., 56 (1997), 279-283.

14. V.G. Pestov, Some properties of free topological groups, Moscow Univ. Math. Bull., 37 (1982), 46-49.

15. I. Protasov, Functionally balanced groups, Math. Notes, 49 (1991), 614-616.

16. I. Protasov, A. Saryev, Semigroups of closed subsets of a topological group, Izv. Akad. Nauk Turkmen SSR, Ser. Fiz.-Tekh. Nauk, 3 (1998), 21-25.

17. M. Tkacenko, On group uniformities on square of a space and extending pseudometrics II, Bull. Austral. Math. Soc., 52 (1995), 41–61.

	On uniformly discrete subsets in uniform spaces and topological groups
Author	T. Banakh, S. Gabriyelyan, I. Protasov 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, Ben-Gurion University of the Negev, Beer-Sheva, 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 third-named author for Polish groups. Applying the obtained results we extend the Hart-van 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.45.1.76-97
Reference	1. A.V. Arhangelskii, M.G. Tkachenko, Topological groups and related strutures, Atlantis Press/World Scientific, Amsterdam-Raris, 2008. 2. W.W. Comfort, S. Hernandez, F.J. Trigos-Arrieta, Relating a locally compact Abelian group to its Bohr compactification, Adv. Math., 120 (1996), 322-344. 3. E.K. van Douwen, The maximal totally bounded group topology an G and the biggest minimal G-space, for abelian groups G, Topology Appl., 34 (1990), 69-91. 4. V. Efremovich, The geometry of proximity, Math. Sb., 73 (1952), 189-200. 5. R. Engelking, General Topology, PWN, Polish Scientific Publ, Warszawa, 1977. 6. L. Gillman, M. Jerison, Rings of continuous functions, Van Nostrand, New York, 1960. 7. I.I. Guran, On topological groups close to being LindelЃNof, Soviet. Math. Dokl., 23 (1981), 173-175. 8. I.I. Guran, On embeddings of topological groups, available in VINITI, Moskow, №1483-81, 1981. 9. K.P. Hart, J. van Mill, Discrete sets and the maximal totally bounded group topology, J. Pure Appl. Algebra, 70 (1991), 73-80. 10. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, V.I, 2nd ed. Springer-Verlag, Berlin, 1979. 11. G. Itzkowitz, Functional balance, discrete balance, and balance in topological groups, Topology Proc., 28 (2004), 569-577. 12. M. Katetov, On real valued functions in topological spaces, Fund. Math., 38 (1951), 85-91. 13. M. Megrelishvili (Levy), P. Nickolas, V. Pestov, Uniformities and uniformly continuous functions on locally connected groups, Bull. Austral. Math. Soc., 56 (1997), 279-283. 14. V.G. Pestov, Some properties of free topological groups, Moscow Univ. Math. Bull., 37 (1982), 46-49. 15. I. Protasov, Functionally balanced groups, Math. Notes, 49 (1991), 614-616. 16. I. Protasov, A. Saryev, Semigroups of closed subsets of a topological group, Izv. Akad. Nauk Turkmen SSR, Ser. Fiz.-Tekh. Nauk, 3 (1998), 21-25. 17. M. Tkacenko, On group uniformities on square of a space and extending pseudometrics II, Bull. Austral. Math. Soc., 52 (1995), 41–61.
Pages	76-97
Volume	45
Issue	1
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On uniformly discrete subsets in uniform spaces and topological groups