Locally uniform spaces

Author:
James Williams

Journal:
Trans. Amer. Math. Soc. **168** (1972), 435-469

MSC:
Primary 54E15

MathSciNet review:
0296891

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Abstract: The axioms for a *locally uniform space* may be obtained by localizing the last axiom for a uniform space to obtain . 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 in which each element of is a neighborhood of the diagonal in 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.

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0296891-5

Article copyright:
© Copyright 1972
American Mathematical Society