On proximal fineness of topological groups in their right uniformity

Ahmed Bouziad


A uniform space X is said to be proximally fine if every proximally continuous function defined on X into an arbitrary uniform pace Y is uniformly continuous. We supply a proof that every topological group which is functionally generated by its precompact subsets is proximally fine with respect to its right uniformity. On the other hand, we show that there are various permutation groups G on the integers N that are not proximally fine with respect to the topology generated by the sets {g ∈ G : g(A) ⊂ B}, A, B ⊂ N.


uniform space; topological group; proximal continuity; proximally fine group; symmetric group; o-radial space

Subject classification

22A05; 54E15.

Full Text:



J. M. Aarts, J. de Groot and R. H. McDowell, Cotopology for metrizable spaces, Duke Math. J. 37 (1970), 291-295. https://doi.org/10.1215/S0012-7094-70-03737-3

A.V. Arkhangel'skii, Topological Function spaces, Vol. 78, Kluwer Academic, Dordrecht, 1992.

A.V. Arhangel'skii, Some properties of radial spaces, Math. Notes Russ. Acad. Sci. 27 (1980), 50-54. https://doi.org/10.1007/BF01149814

T. Banakh, I. Guran and I. Protasov, Algebraically determined topologies on permutation groups, Topology Appl. 159 (2012), 2258-2268. https://doi.org/10.1016/j.topol.2012.04.010

A. Bouziad and J.-P. Troallic, Problems about the uniform structures of topological groups, in: Open Problems in Topology II. Ed. Elliott Pearl. Amsterdam: Elsevier, 2007, 359-366. https://doi.org/10.1016/B978-044452208-5/50038-7

A. Bouziad and J.-P. Troallic, Left and right uniform structures on functionally balanced groups, Topology Appl. 153, no. 13 (2006), 2351-2361. https://doi.org/10.1016/j.topol.2005.03.017

D. Dikranjan and A. Giordano Bruno, Arnautov's problems on semitopological isomorphisms, Appl. Gen. Topol. 10, no. 1 (2009), 85-119. https://doi.org/10.4995/agt.2009.1789

R. Engelking, General Topology, Heldermann, Berlin, 1989.

H. Fuhr and W. Roelcke, Contributions to the theory of boundedness in uniform spaces and topological groups, Note di Matematica 16, no. 2 (1996), 189-226.

S. Hartman and J. Mycielski, On the imbedding of topological groups into connected gopological groups, Colloq. Math. 5 (1958), 167-169. https://doi.org/10.4064/cm-5-2-167-169

H. Herrlich, Quotienten geordneter Raume und Folgenkonvergenz, Fund. Math. 61 (1967), 79-81. https://doi.org/10.4064/fm-61-1-79-81

M. Husek, Ordered sets as uniformities, Topol. Algebra Appl. 6, no. 1 (2018), 67-76. https://doi.org/10.1515/taa-2018-0007

G. L. Itzkowitz, Continuous measures, Baire category, and uniform continuity in topological groups, Pacific J. Math. 54 (1974), 115-125. https://doi.org/10.2140/pjm.1974.54.115

M. Katetov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85-91. https://doi.org/10.4064/fm-38-1-85-91

I. V. Protasov, Functionally balanced groups, Mat. Zametki 49, no. 6 (1991), 87-91, Translation: Math. Notes 49, no. 6 (1991) , 614-616. https://doi.org/10.1007/BF01156586

W. Roelcke and S. Dierolof, Uniform structures in topological groups and their quotients, McGraw-Hill, New York, 1981.

M. Shlossberg, Balanced and functionally balanced P-groups, Topol. Algebra Appl. 6, no. 1 (2018), 53-59. https://doi.org/10.1515/taa-2018-0006

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