On separation axioms of uniform bundles and sheaves

Abstract

In the context of the theory of uniform bundles in the sense of J. Dauns and K. H. Hofmann, the topology of the fiber space of a uniform bundle depends on the assumption of upper semicontinuity of its defining set of pseudometrics when composed with local sections. In this paper we show that the additional hypothesis of lower semicontinuity of these functions secures that the fiber space of the uniform bundle is Hausdorff, regular or completely regular provided that the base space has the corresponding separation axiom. Similar results for the particular important case of sheaves of sets follow suit.

Keywords

• Uniform bundle
• Sheaf of sets
• Lower semicontinuity
• Upper semicontinuity
• Separation axioms