UNIFORM PROPERTIES AND HYPERSPACE TOPOLOGIES FOR ALEPH-UNIFORMITIES

Let X be a completely regular space and aleph an infinite cardinal number. The aleph-uniformity of X generated by all open normal coverings of X with cardinality less-than-or-equal-to aleph is the weakest one with the following property: any continuous function from X to any metric space of weight less-than-or-equal-to aleph is uniformly continuous. Any continuous function from a uniform space X to any metric space of weight less-than-or-equal-to aleph is uniformly continuous iff any locally finite covering of cozero-sets of cardinality less-than-or-equal-to aleph is uniform. With aleph-collectionwise normality, any continuous function from X to any metric space of weight less-than-or-equal-to aleph and uniform dimension less-than-or-equal-to 1 is uniformly continuous iff any discrete family of subsets of X with cardinality less-than-or-equal-to aleph is uniformly discrete. The uniform hypertopologies induced via the Hausdorff uniformity on the hyperspace 2X of X from the aleph-uniformity, generated by the family of all continuous functions from X to any metric space of density less-than-or-equal-to aleph and uniform dimension less-than-or-equal-to 1 and from the aleph-uniformity agree. Further, both agree with a Vietoris-type topology iff X is normal.