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Title:
Axiomatic analysis via value quantales
Untitled
Author:
Chand, Alveen Aditya
Institution:
University of the South Pacific.
Award:
M.Sc. Mathematics
Subject:
Lattice theory, Ordered algebraic structures, Axiomatic set theory
Date:
2014
Call No.:
Pac QA 171 .5 .C53 2014
BRN:
1197593
Copyright:
20-40% of this thesis may be copied without the authors written permission
Abstract:
Since the introduction of the notion of metric space, mathematics have shown immense interest towards it. Metric spaces represent an abstraction of the notion of distance, thus significant proportion of the research throughout the decades has been focused on its axioms. This thesis discusses a systematic analysis of quantale valued metric spaces (or V spaces), a generalization of metric spaces which uses value quantales. A comparative study of basic results in Metric Analysis is conducted in regards to value quantales. Some significant results such as continuity of the binary operations on V-spaces are established along with results on sequential continuity, convergence of monotone nets and a generalized Squeeze Lemma. Some of these results enlighten on the importance of the properties of real numbers and assist in categorizing the properties as either general or special . Also, an example of the value quantale of distance distribution function is presented in detail; such an example is harly shown in existing literature. Furthermore, a construction of the completion of V-spaces which satisfy uniformly vanishing asymmetry (a weak form of sysmmetry) is given using Cauchy filters. This construction along with other results on V-spaces establish that quantale valued metric spaces give a powerful axiomatization of Analysis.
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