Uniform properties and hyperspace topologies for ℵ-uniformities

Abstract

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