Minimal topological completion of KBan1→KVec

1982 ◽  
pp. 123-132 ◽  
Author(s):  
Rudolf-E. Hoffmann
Keyword(s):  
Topological Completion

scholarly journals p-topological Cauchy completions

1999 ◽  
Vol 22 (3) ◽  
pp. 497-509
Author(s):  
J. Wig ◽  
D. C. Kent
Keyword(s):  
Equivalence Class ◽  
Topological Property ◽  
Convergence Space ◽  
Extension Theorems ◽  
General Properties ◽  
Continuous Map ◽  
Cauchy Space ◽  
Topological Completion ◽  
Cauchy Spaces ◽  
Natural Way

The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular,p-regular, and topological Cauchy completions, we hereby initiate a study ofp-topological Cauchy completions. Ap-topological Cauchy space has ap-topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing ap-topological completion, it is shown that a certain class of Reed completions preserve thep-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsestp-topological completions. However, not allp-topological completions are Reed completions. Several extension theorems forp-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowingp-topological andp′-topological completions, respectively, can always be extended to aθ-continuous map between anyp-topological completion of the first space and anyp′-topological completion of the second.

scholarly journals Characterizing the continuous functionals

1983 ◽  
Vol 48 (4) ◽  
pp. 965-969 ◽  
Author(s):  
Dag Normann
Keyword(s):  
Real Line ◽  
Natural Extension ◽  
Rational Numbers ◽  
The Real ◽  
Finite Systems ◽  
Several Variables ◽  
Finite Structures ◽  
Starting Point ◽  
Topological Completion ◽  
Continuous Functionals

One of the objectives of mathematics is to construct suitable models for practical or theoretical phenomena and to explore the mathematical richness of such models. This enables other scientists to obtain a better understanding of such phenomena. As an example we will mention the real line and related structures. The line can be used profitably in the study of discrete phenomena like population growth, chemical reactions, etc.Today's version of the real line is a topological completion of the rational numbers. This is so because then mathematicians have been able to work out a powerful analysis of the line. By using the real line to construct models for finitary phenomena we are more able to study those phenomena than we would have been sticking only to true-to-nature but finite structures.So we may say that the line is a mathematical model for certain finite structures. This motivates us to seek natural models for other types of finite structures, and it is natural to look for models that in some sense are complete.In this paper our starting point will be finite systems of finite operators. For the sake of simplicity we assume that they all are operators of one variable and that all the values are natural numbers. There is a natural extension of the systems such that they accept several variables and give finite operators as values, but the notational complexity will then obscure the idea of the construction.

Generalization of Hölder's Theorem to Ordered Modules

1969 ◽  
Vol 21 ◽  
pp. 149-157 ◽  
Author(s):  
T. M. Viswanathan
Keyword(s):  
Sufficient Conditions ◽  
Open Interval ◽  
Order Structure ◽  
Real Numbers ◽  
Ordered Group ◽  
Quotient Field ◽  
Interval Topology ◽  
Ordered Field ◽  
Topological Completion ◽  
Necessary And Sufficient

Hölder's theorem on archimedean groups states:An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

scholarly journals The topological completion of a bilinear form

2002 ◽  
Vol 118 (3) ◽  
pp. 337-344
Author(s):  
Solomon Jekel ◽  
Neal Macmillan
Keyword(s):  
Bilinear Form ◽  
Topological Completion

scholarly journals On the topological completion

1976 ◽  
Vol 56 (1) ◽  
pp. 365-365 ◽  
Author(s):  
Howard Curzer ◽  
Anthony W. Hager
Keyword(s):  
Topological Completion
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