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Journal Articles Topology and its Applications Year : 2009

Uniformities with the same Hausdorff hypertopology

Ahmed Bouziad
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Abstract

Two uniformities U and V on a set $X$ are said to be H-equivalent if their corresponding Hausdorff uniformities on the set of all non-empty subsets of $X$ induce the same topology. The uniformity U is said to be H-singular if no distinct uniformity on $X$ is H-equivalent to U. The self-explanatory concepts of H-coarse, H-minimal and H-maximal uniformities are defined similarly. It is well known that not all uniformities are H-singular. We show here that there is a property which obstructs H-singularity: Every H-minimal uniformity has a base of finite-dimensional uniform coverings. Besides, we provide an intrinsic characterization of H-minimal uniformities and show that they are H-coarse. This characterization of H-minimality becomes a criterion for H-singularity for all uniformities that are either complete, uniformly locally precompact or proximally fine (e.g., metrizable ones). Some relevant properties which insure H-singularity are introduced and investigated in some aspect.
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Dates and versions

hal-00372922 , version 1 (02-04-2009)

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  • HAL Id : hal-00372922 , version 1

Cite

Ahmed Bouziad. Uniformities with the same Hausdorff hypertopology. Topology and its Applications, 2009, Volume 156, Issue 7 (1), pp.1315-1326. ⟨hal-00372922⟩
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