On Fuzzy Compactness

Abstract

The fundamental concept of a fuzzy set and fuzzy set operations was first introduced by L. A. Zadeh [175] in 1965 and it provides a natural foundation for treating mathematically the fuzzy phenomena, which exists pervasively in our real world and for building new branches of fuzzy mathematics. This also provides a natural frame work for generalizing various branches of mathematics such as fuzzy topology, fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy supra topology, fuzzy infra topology, fuzzy bitopology etc. C. L. Chang [19] in 1968 first introduced the concept of fuzzy topological spaces by using fuzzy sets. C. K. Wong [160, 161, 162, 162], R. Lowen [107, 108, 109, 110,111], B. Hutton [70, 71, 72], T. E. Gantner et al. [54], P. P. Ming and L. Y. Ming [121, 122], etc., discussed various aspects of fuzzy topology by using fuzzy sets. A. J. Klein [91] defines  -level sets and  -level topology. Fuzzy compactness occupies a very important place in fuzzy topological spaces and so does some of its forms. Fuzzy compactness first discussed by C. L. Chang [19], T. E. Gantner et al. [54] introduced  -compactness, A. D. Concilio and G. Gerla [27] discussed almost compact spaces and M. N. Mukherjee and A. Bhattacharyya [130] discussed almost  -compact spaces. The purpose of this thesis is to contribute about different types of fuzzy compactness and establish theorems, corollaries and examples in fuzzy topological spaces by using the definitions of C. L. Chang [19], T. E. Gantner et al. [54], A. D. Concilio and G. Gerla [27] and M. N. Mukherjee and A. Bhattacharyya [130]. We study several properties of these definitions along with the different theorems from existing there. Moreover to suggest new definitions of fuzzy  -compact spaces,  -compact fuzzy sets,  - -compact spaces, partially  -compact and partially  - -compact fuzzy sets, Q-compact and  -Q-compact fuzzy sets, Q -compact and  -Q -compact fuzzy sets, almost partially  -compact and almost partially  - -compact fuzzy sets, almost Q -compact and almost  -Q -compact fuzzy sets and also to study their several properties in fuzzy topological spaces have been done in the work. Chapter one incorporates some fundamental definitions and results of fuzzy sets, fuzzy set operations, fuzzy mapping, fuzzy topology, fuzzy separation axioms, good extension property and fuzzy productivity. These results are ready bibliographies for the study in the next chapters. Results are stated without proof and can be found in the thesis referred to. Our works start from chapter two. Chapter two deals with fuzzy compact spaces due to C. L Chang [19] which is global property. In this chapter, we have discussed some theorems, corollaries and examples in fuzzy topological spaces, fuzzy subspaces, mappings in fuzzy topological spaces, fuzzy 1 T -spaces, fuzzy Hausdorff spaces, fuzzy regular spaces and good extension property about fuzzy compact spaces. Also we have defined  -open fuzzy sets,  -cover, fuzzy  -compact spaces and investigated difference between fuzzy compact and fuzzy  -compact spaces. We aim to study  -compact spaces in the sense of T. E. Gantner et al. [54] in chapter three which is global property and we have introduced  -level continuous mapping. In this chapter, we have established some theorems, corollaries and examples in fuzzy topological spaces, fuzzy subspaces, mappings in fuzzy topological spaces, fuzzy 1 T -spaces, fuzzy Hausdorff spaces, fuzzy regular spaces,  -level topological spaces, cofinite topological spaces, good extension property and fuzzy product spaces and give some examples about  -compact spaces. Also we have constructed  - -shading,  - -compact spaces and identified difference between  -compact and  - -compact spaces. We have discussed compact fuzzy sets due to C. L. Chang [19] in chapter four which is local property. In this chapter, we have investigated some theorems, corollaries and examples of compact fuzzy sets in fuzzy topological spaces, fuzzy subspaces, fuzzy mappings, fuzzy 1 T -spaces, fuzzy Hausdorff spaces, fuzzy regular spaces, good extension property and fuzzy productivity about compact fuzzy sets. Also we have introduced  -compact fuzzy sets and found difference between compact and  -compact fuzzy sets. In chapter five, we have defined partial  -shading, partial  -subshading, open partial  -shading, partially  -compact fuzzy sets. We have discussed some theorems, corollaries and examples of partially  -compact fuzzy sets in fuzzy topological spaces, fuzzy subspaces, fuzzy mappings,  -level continuous mapping, fuzzy 1 T -spaces, fuzzy Hausdorff spaces, fuzzy regular spaces,  -level topological spaces, good extension property and fuzzy productivity about partially  -compact fuzzy sets. Also we have introduced partial  - -shading, partial  - -subshading and partially  - -compact fuzzy sets and indicated the difference between partially  -compact and partially  - -compact fuzzy sets. In chapter six, we have constructed Q-cover, Q-subcover, open Q-cover, Q-compact fuzzy sets, Q -cover, Q -subcover, open Q -cover, Q -compact fuzzy sets,  -Q-cover,  -Q-subcover,  -Q-compact fuzzy sets,  -Q -compact fuzzy sets. We have also studied some theorems, corollaries and examples in fuzzy topological spaces, fuzzy subspaces, fuzzy 1 T -spaces, fuzzy Hausdorff spaces, fuzzy regular spaces,  -level topological spaces, good extension property and fuzzy productivity about Q-compact, Q -compact,  -Q-compact,  -Q -compact fuzzy sets. Furthermore, we have found difference between Q-compact and Q -compact fuzzy sets, Q-compact and  -Q-compact fuzzy sets, Q -compact and  -Q -compact fuzzy sets. Moreover, we have compared compact fuzzy stes (Chang’s sense [19]) with Q-compact and Q -compact fuzzy sets,  -compact fuzzy stes (Chang’s sense [19]) with  -Q-compact and  -Q -compact fuzzy sets. In chapter seven, we have studied almost compact fuzzy sets due to A. D. Concilio and G. Gerla [27] which is local property. We have established some theorems, corollary and give some examples in fuzzy topological spaces, fuzzy subspaces, fuzzy mappings, fuzzy 1 T -spaces, fuzzy regular spaces, good extension property and fuzzy productivity about almost compact fuzzy sets. Also we have introduced proximate  -cover, proximate  -subcover, almost  -compact fuzzy sets and found different characterizations between almost compact and almost  -compact fuzzy sets. We have dealth with almost  -compact spaces due to M. N. Mukherjee and A. Bhattacharyya [130] in chapter eight which is global property. In this chapter, we have established some theorems, corollary and give some examples in fuzzy topological spaces, fuzzy subspaces, fuzzy mappings, fuzzy 1 T -spaces, fuzzy regular spaces,  -level topological spaces,  -level continuous mapping and good extension property about almost  -compact spaces. Also we have introduced proximate  - -shading, proximate  - -subshading, almost  - -compact spaces and found difference between almost  -compact and almost  - -compact spaces. In chapter nine, we have introduced proximate partial  -shading, proximate partial  -subshading, almost partially  -compact fuzzy sets. We have also established some theorems, corollary and give some examples in fuzzy topological spaces, fuzzy subspaces, fuzzy mappings, fuzzy 1 T -spaces, fuzzy regular spaces,  -level topological spaces,  -level continuous mapping, good extension property and fuzzy productivity about almost partially  -compact fuzzy sets. In addition to that, we have defined proximate partial  - -shading, proximate partial  - -subshading, almost partially  - -compact fuzzy sets and investigated different characterizations between almost partially  -compact and almost partially  - -compact fuzzy sets. In chapter ten, we have defined proximate Q -cover, proximate Q -subcover, almost Q -compact fuzzy sets. We have also studied some theorems, corollary and give some examples in fuzzy topological spaces, fuzzy subspaces, fuzzy 1 T -spaces, fuzzy regular spaces,  -level topological spaces, good extension property and fuzzy productivity about almost Q -compact fuzzy sets. Moreover, we have introduced proximate  -Q -cover, proximate  -Q -subcover, almost  -Q -compact fuzzy sets and found different characterizations between almost Q -compact and almost  -Q -compact fuzzy sets.