Uniformities with the same Hausdorff hypertopology

AbstractTwo 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