On proximal fineness of topological groups in their right uniformity

Ahmed Bouziad

Abstract

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.


Keywords

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

Subject classification

22A05; 54E15.

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References

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