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Generalisations of Filters and Uniform Spaces

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Generalisations of Filters and Uniform Spaces Rhodes University Department Of Mathematics Generalisations of Filters and Uniform Spaces Murugiah Muraleetharan A thesis submitted in ful¯lment of the requirements for the Degree of Master of Science In Mathematics Abstract X X The notion of a ¯lter F 2 22 has been extended to that of a : pre¯lter F 2 2I , X X generalised ¯lter f 2 I2 and fuzzy ¯lter ' 2 II . A uniformity is a ¯lter with some X£X other conditions and the notion of a uniformity D 2 22 has been extended to that of X£X X£X a : fuzzy uniformity D 2 2I , generalised uniformity d 2 I2 and super uniformity X£X ± 2 II . We establish categorical embeddings from the category of uniform spaces into the categories of fuzzy uniform spaces, generalised uniform spaces and super uniform spaces and also categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces. KEYWORDS: Pre¯ter, Generalised ¯lter, Fuzzy ¯lter, Fuzzy uniform space, Generalised uniform space, Super uniform space, Embedding functor and Functor isomorphim. A. M. S. SUBJECT CLASSIFICATION: 03E72, 04A20, 18A22, 54A40, 54E15. Contents 1 Fuzzy Sets 1 1.1 Introduction . 1 1.2 Crisp Subsets of X Associated With a Fuzzy Set . 3 1.3 Fuzzy Sets Induced by Maps . 6 2 Fuzzy Topology 9 2.1 De¯nitions and Fundamental Properties . 9 2.2 Fuzzy Closure Operator . 11 2.3 Continuous Functions . 12 3 Filters 14 3.1 Introduction . 14 3.2 Ultra¯lters . 15 3.3 Topological Notions in Terms of Filters . 18 4 Uniform Spaces 21 4.1 Introduction . 21 4.2 The Uniform Topology . 23 4.3 Uniformly Continuous Functions . 24 5 Pre¯lters 26 5.1 Introduction . 26 5.2 Pre¯lters from Filters . 33 5.3 Filters from Pre¯lters . 35 5.4 Images and Preimages . 43 5.5 Convergence in Fuzzy Topological Space . 44 6 Fuzzy Uniform Spaces 47 6.1 Introduction . 47 6.2 Fuzzy Neighbourhood Spaces . 49 6.3 Fuzzy Uniform Topology . 52 6.4 Convergence in Fuzzy Uniform Topology . 58 6.5 Uniformly Continuous Functions . 59 6.6 The ®-Level Uniformities . 61 7 Generalised Filters 67 7.1 Introduction . 67 7.2 G-¯lters from Pre¯lters . 71 7.3 Pre¯lters from G-¯lters . 72 7.4 Prime G-Filters . 78 8 Generalised Uniform Spaces 83 8.1 Introduction . 83 8.2 Uniformly Continuous Functions . 86 1 9 Fuzzy Filters 88 9.1 Inroduction . 88 9.2 Fuzzy Filters with Characteristic 1 . 92 9.3 Fuzzy Filters From G-Filters and G-Filters From Fuzzy Filters . 97 10 Super Uniform Spaces 100 10.1 Introduction . 100 10.2 Uniformly Continuous Functions . 105 11 Categorical Embeddings 108 11.1 Embedding into the Category of Fuzzy Uniform Spaces . 108 11.2 Embedding into the Category of Generalised Uniform Spaces . 109 11.3 Embeddings into the Category of Super Uniform Spaces . 111 2 ACKNOWLEDGEMENTS In presenting this thesis, the result of 18 months of study within the mathematics de- partment of Rhodes University, I wish to express my warm thanks to those who have given me assistance and encouragement during my time here. First of all, I am indebted to the Head of Department Professor Wesley Kotze. I am grateful to him for his guidance and for the ¯nancial assistance he has provided through the FRD without which I could not have completed the course. I also wish to place on record the very helpful support of my supervisor Dr. Mike Burton. Such progress as I have made in the present enquiry owes much to the useful and regular discussions I have had with him. I am also indebted to my colleague Mr Alan Roy, a graduate of the University of Aberdeen in Scotland whose friendship has meant much to me. The peaceful orderly environment of Rhodes University coupled with the friendly atmosphere of the department I have been privileged to enjoy. Above all, I would like to thank my family and in particular my father who was an inspiring mathematics teacher. From a young age his help and encouragement have proved invaluable in the development of my career as a mathematician. 3 PREFACE Since the notion of a fuzzy set was introduced by Zadeh [84], there have been attempts to extend useful mathematical notions to this wider setting, replacing sets by fuzzy sets. The study of uniform space notion is facilitated by the notion of a ¯lter. In [52] Lowen introduced the notion of a fuzzy uniform space. This was accomplished with the aid of a ¯lters on IX which Lowen [48] called pre¯lters. The notions of pre¯lter and fuzzy uniform space are extentions of ¯lter and uniform space. We prove that the category of uniform spaces is embedded into the category of fuzzy uniform spaces. The notion of a fuzzy ¯lter is introduced in [22] and then a new generalised form of uniformity, called super uniformity, is de¯ned and studied. It is based on the concept of fuzzy ¯lter. In [15] the notion of a generalised uniform space is introduced and studied using pre¯lters, which are not the most natural analogue of the ¯lter notion in this situation. With the collaboration of Burton and Guti¶errezwe introduced and studied the notion of a generalised ¯lter in [16]. In Chapter 8, we rewrite the basic theory of generalised uniform spaces with the aid of generalised ¯lters. We can see that the notion of a generalised uniform space and the notion of a super uniform space are extentions of uniform space and the category of uniform spaces is embed- ded into the categories of generalised uniform spaces and super uniform spaces. We prove that the category of generalised uniform spaces and the category of fuzzy uniform spaces are isomorphic. We also show that the categories of fuzzy uniform spaces and generalised unifrom spaces are embedded into the category of super uniform spaces. We ¯rst establish consistent notation. We have also slightly changed a few de¯nitions and strengthened some theorems. This lead us to establish a nice categorical connection between the categories of uniform spaces, fuzzy uniform spaces, generalised uniform spaces and super uniform spaces. Chapters 1 and 2 introduce basic results in fuzzy sets and fuzzy topology. Chapter 1 gives the introduction to fuzzy sets and deals with crisp subsets associated with a fuzzy set and fuzzy sets induced by maps. The second chapter is concerned with fuzzy topology: the fuzzy closure operator is a very useful tool to de¯ne a topology. We also see in this chapter continuous functions between fuzzy uniform spaces. In chapters 3 and 4 we record the standard results on ¯lters and uniform spaces, since they give and an idea for consequent results in later chapters. Chapter 5 explores the fundamental ideas of pre¯ters. We get pre¯lters from ¯lters and ¯lters from pre¯lters. We see the images and preimages of ¯lters under a map and study convergence in fuzzy topological space. In chapter 6 we de¯ned fuzzy topology as a pre¯lter plus other conditions. We deal some basic results in fuzzy neighbourhood spaces which are essential to ¯nd a fuzzy closure operator and then obtain a fuzzy topology from it. We ¯nd fuzzy topologies from fuzzy uniform space directly and from fuzzy neighbourhood spaces. Next we deal with convergence in uniform topology and uniformly continuous functions between fuzzy uniform spaces. In the last section we deal with ®-level uniformities which turned out to be very useful for subsequent results. Chapter 7 is concened with generalised ¯lters: de¯nitions, generalised ¯lters from pre- ¯lters and pre¯lters from generalised ¯lters and prime generalised ¯lters. In Chapter 8 we deal with generalised uniform spaces and uniformly continuous functions between generalised uniform spaces. We will see fuzzy ¯lters in Chapter 10 with more emphasis given to a fuzzy ¯lter with characteristic value 1. We also obtain fuzzy ¯lters from generalised ¯lters and generalised ¯lters from fuzzy ¯lters. In Chapter 10 we give de¯nitions for ®-uniformities and super uniformities. Then we see uniformly continuous functions between ®-uniform spaces and super uniform spaces. Chapter 11 is the centeral chapter which connects the categories of uniform spaces, fuzzy 4 uniform spaces, generalised uniform spaces and super uniform spaces. Here we see that fuzzy uniform spaces, generalised uniform spaces and super uniform spaces are extentions of uniform space using categorical embeddings from the category of uniform spaces into these categories. We can also obtain categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces. Also we show that the category of fuzzy uniform spaces and the category of generalised uniform spaces are isomorphic. 5 Chapter 1 Fuzzy Sets 1.1 Introduction The role of set theory has been formulated in the development of modern mathematics. However usual observables in our daily lives and conversation as well as scienti¯c experi- mentation constitute ill-de¯ned sets. For example, the set of old people, the class of tall men, the class of large numbers, the set of low temperatures etc. In order to try to develop a theory for such ill-de¯ned sets L.A. Zadeh [84] de¯ned the notion of a fuzzy set as follows. 1.1.1 De¯nition Let X be a set. A fuzzy set on X is a map from X into [0; 1]. That is, if ¹ is a fuzzy set on X then ¹ 2 IX . Where I = [0; 1] and IX denotes the collection of all maps from X into I.
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