Locally uniform spaces

Abstract

The axioms for a locally uniform space ( X , U ) (X,\mathcal {U}) may be obtained by localizing the last axiom for a uniform space to obtain ∀ x ∈ X , ∀ U ∈ U , ∃ V ∈ V : ( V ∘ V ) [ x ] ⊆ U [ x ] \forall x \in X,\forall U \in \mathcal {U},\exists V \in \mathcal {V}:(V \circ V)[x] \subseteq U[x] . With each locally uniform space one may associate a regular topology, just as one associates a completely regular topology with each uniform space. The topologies of locally uniform spaces with nested bases may be characterized using Boolean algebras of regular open sets. As a special case, one has that locally uniform spaces with countable bases have pseudo-metrizable topologies. Several types of Cauchy filters may be defined for locally uniform spaces, and a major portion of the paper is devoted to a study and comparison of their properties. For each given type of Cauchy filter, complete spaces are those in which every Cauchy filter converges; to complete a space is to embed it as a dense subspace in a complete space. In discussing these concepts, it is convenient to make the mild restriction of considering only those locally uniform spaces ( X , V ) (X,\mathcal {V}) in which each element of V \mathcal {V} is a neighborhood of the diagonal in X × X X \times X with respect to the relative topology; these spaces I have called NLU-spaces. With respect to the more general types of Cauchy filters, some NLU-spaces are not completable; this happens even though some completable NLU-spaces can still have topologies which are not completely regular. Examples illustrating these completeness situations and having various topological properties are obtained from a generalized construction. It is also shown that there is a largest class of Cauchy filters with respect to which each NLU-space has a completion that is also an NLU-space.