Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Locally uniform spaces
HTML articles powered by AMS MathViewer

by James Williams PDF
Trans. Amer. Math. Soc. 168 (1972), 435-469 Request permission

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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 54E15
  • Retrieve articles in all journals with MSC: 54E15
Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 168 (1972), 435-469
  • MSC: Primary 54E15
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0296891-5
  • MathSciNet review: 0296891