The Study Of Linear Fuzzifying Uniform Spaces And Bornological Linear Spaces

As a link between metric spaces and topological spaces,uniform spaces have been received widely concerned by many researchers in studying the theory of fuzzy topology.On the other hand,in recent years,bornological spaces have become a hot topic because this kind of spaces can be used as an ideal tool to study noncommutative geometry and representation theory,global analysis and optimization theory.How to deal with the uncertain objects in the framework of bornological spaces has become a frontier problem in the theory of fuzzy analysis,in which the first choice is to carry out the research of fuzzifying bornological linear spaces based on fuzzy logic.This thesis attempts to study the compatibility of Hutton type fuzzifying uniformities and linear operation,fuzzifying bornological linear spaces and related problems in the framework of fuzzy logic systematically.This thesis mainly contains two parts.The Hutton type linear fuzzifying uniform spaces are introduced and some related properties are investigated,such as completely boundedness,Hausdorff separation and their relationships in the first part.Also,the initial and final Hutton type linear fuzzifying uniformities are studied.In the second part,the fuzzifying bornological linear spaces are discussed.The hornologically convergence and separation in fuzzifying bornological linear spaces are also studied.In addition,the notions of fuzzifying bornological filters and net convergence are investigated,respectively.Moreover,some properties are discussed in fuzzifying bornological linear spaces such as completeness,compactness and precompactness.The relationships between these properties in fuzzifying bornological linear spaces are also studied.Finally,the notion of L-bornological linear spaces is introduced and the induced L-bornological linear space is also obtained.In the following,let me explain explicitly what I have done.In Chapter 1,it shows the research status of related content at home and abroad and the significance of this thesis.In Chapter 2,some necessary notions and fundamental results which are used in this thesis about fuzzifying uniformities and fuzziying bornologies are recalled.In Chapter 3,the main purpose is to study Hutton type fuzzifying uniformities on linear spaces.Firstly,we show that if a base of a fuzzifying uniformity defined over a linear space is translation-invariant,balanced and absorbed,then it can generate a linear fuzzifying topology.From this generated linear fuzzifying topology,we can construct a new linear fuzzifying uniformity(i.e.,a fuzzifying uniformity compatible with the linear structure)which is equivalent to the original fuzzifying uniformity.Secondly,the Hausdorff separation and complete boundedness in linear fuzzifying uniformitis are investigated.As an example,the linear fuzzifying uniformity induced by a fuzzy norm is also discussed.In addition,it is also shown the specific characterization of initial and final Hutton type linear fuzzifying uniformities.In Chapter 4,the notion of a fuzzifying bornological linear space is introduced and some examples of fuzzifying bornological linear spaces are provided,such as the fuzzifying linear bornology induced by probabilistic precompact sets of a Menger PNspace.The concrete illustration of fuzzifying bornological convergence is shown and some properties of fuzzifying bornological convergence in fuzzifying bornological linear spaces are discussed.After considering the specific description of fuzzifying bornological close,the separation of fuzzifying bornological linear spaces is investigated as well as product and quotient fuzzifying bornological linear spaces.Additionally,the notion of fuzzifying bornological filter convergence is introduced.The relationship between fuzzifying bornological filter and net convergence are shown.Furthermore,some properties are discussed in fuzzifying bornological linear spaces such as completeness,compactness and precompactness.Particularly,it is demonstrated that a subset is bornological compact if and only if it is bornological precompact and bornological complete.In Chapter 5,the specific description of L-bornological linear spaces is presented,some properties of Lowen functors between the category of bornological linear spaces and the category of L-bornological linear spaces are discussed.In addition,the notions and some properties of L-Mackey convergence and separation in L-bornological linear spaces are showed.In particular,the characterization of separation in L-bornological linear spaces in terms of L-Mackey convergence is obtained.In Chapter 6,the main contribution of this thesis is summarized and the expection are made.