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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A class of uniform convergence structures

Author: G. D. Richardson
Journal: Proc. Amer. Math. Soc. 25 (1970), 399-402
MSC: Primary 54.22
MathSciNet review: 0256335
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Abstract: In 1967, Cook and Fischer introduced in the journal Mathematische Annalen the notion of a uniform convergence structure, abbreviated u.c.s., for a set $ X$. Here we consider the class $ \Gamma $ of u.c.s. which have the following property: a u.c.s. $ I \in \Gamma $ provided there is a filter $ \Phi \in I$ such that $ \mathcal{F}$ is finer than $ \Phi (x)$ for every filter $ \mathcal{F}$ which converges to $ x$, for each $ x \in X$. Various properties of the class $ \Gamma $ are discussed. The main result is that a topology $ \tau $ for $ X$ is regular if and only if there is an $ I \in \Gamma $ such that $ I$ induces $ \tau $. Also it it is shown that each $ I \in \Gamma $ induces a regular topology for $ X$.

The class $ {\Gamma _0}$ of u.c.s. which satisfy the completion axiom was first introduced by Biesterfeldt, Indag. Math., 1966. Here it is shown that $ {\Gamma _0} \subset \Gamma $ and a characterization of the class $ {\Gamma _0}$ is given in terms of Cauchy filters.

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Keywords: Uniform convergence structures, symmetric filters, diagonal filters, ultrafilters, Cauchy filters, regular topologies
Article copyright: © Copyright 1970 American Mathematical Society