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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $ (X,\mathcal{U})$ may be obtained by localizing the last axiom for a uniform space to obtain $ \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,\mathcal{V})$ in which each element of $ \mathcal{V}$ is a neighborhood of the diagonal in $ 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.

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Article copyright: © Copyright 1972 American Mathematical Society