A class of uniform convergence structures
Author:
G. D. Richardson
Journal:
Proc. Amer. Math. Soc. 25 (1970), 399-402
MSC:
Primary 54.22
DOI:
https://doi.org/10.1090/S0002-9939-1970-0256335-X
MathSciNet review:
0256335
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Abstract | References | Similar Articles | Additional Information
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.
- H. J. Biesterfeldt Jr., Completion of a class of uniform convergence spaces, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28 (1966), 602–604. MR 0205219 N. Bourbaki, General topology. Part I, Hermann, Paris and Addison-Wesley, Reading, Mass., 1966. MR 34 #5044a.
- C. H. Cook and H. R. Fischer, Uniform convergence structures, Math. Ann. 173 (1967), 290–306. MR 217756, DOI https://doi.org/10.1007/BF01781969
- H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269–303 (German). MR 109339, DOI https://doi.org/10.1007/BF01360965
- D. C. Kent, A note on pretopologies, Fund. Math. 62 (1968), 95–100. MR 224055, DOI https://doi.org/10.4064/fm-62-1-95-100
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Additional Information
Keywords:
Uniform convergence structures,
symmetric filters,
diagonal filters,
ultrafilters,
Cauchy filters,
regular topologies
Article copyright:
© Copyright 1970
American Mathematical Society