On the use of partial orders in uniform spaces

Abstract

We investigate the use of nets indexed by preorders in uniform spaces. Nine different Cauchy conditions and four different convergence conditions yield 36 completeness properties, each of which turns out to be equivalent to a known form of completeness. We also use these preordered nets to characterize the functors θ, λ, and v, which are associated with these completeness properties. In the case of λ we give an example to show that the analogous characterization with predirected nets does not work.

Keywords

• Locally fine
• Complete
• Supercomplete
• Cofinally complete
• Paracompact