Diagonal Cauchy Spaces D.C

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Uniformizable Cauchy Spaces

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Completion of a Cauchy Space Without the T2-Restriction on the Space

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Graph Minor from Wikipedia, the Free Encyclopedia Contents

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A Completion Functor for Cauchy Groups

I nternat. J. Math. Math. Si. 55 Vol. 4 No. (1981)55-65 A COMPLETION FUNCTOR FOR CAUCHY GROUPS R. FRIC D.C. KENT Department of Pure and Applied Mathematics Washington State University Pullman, Washington 99164 U.S.A. (Received March ii, 1980) ABSTRACT. A completion functor is constructed on the category of completely normal Cauchy groups and Cauchy-continuous homomorphlsms. A competlon functor is also obtained for a corresponding category of convergence groups. KEY WORDS AND PHRASES. Completion functor, Cauchy group, pre-Cauchy group, completely normal Cauchy group, convergence group 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 54A20, 54E15, 18A5 i. INTRODUCTION. In [ 2] we developed a general functorial approach to Cauchy space com- pletions. In this paper, we consider completions of groups equipped with com- patible Cauchy structures. Such spaces have been studied by R.N. Ball [ I], who used them to obtain certain completions of lattice ordered groups. We shall use the definitions, notation, and terminology of [ 2]. However the modification functor approach to constructing completion functors used in 56 R. FRIC AND D.C. KENT [2] is not employed here. Although at least some of our results could be obtained via that method, it is found to be less convenient due, in part, to the fact that Wyler's completion does not preserve the compatibility between group and Cauchy structures. Let PCHG be the category whose objects are groups equipped with Cauchy strudtures relative to which the group operations are Cauchy continuous (such groups are said to be pre-Cauchy), and whose morphisms are Cauchy-continuous homomorphisms. All Cauchy space categories considered here will be subcategories of PCHG.