WEAK FORMS OF SOME NOTIONS IN FUZZY TOPOLOGY

Abstract

N. Levine [33] introduced the concepts of semi-open sets and semi-continuous mappings in topological spaces. Thereafter many researchers contributed to this area: Andrijevic [7] studied semi-preopen sets in 1986. D.E. Cameron and G. Woods [12] studied the notions of s-continuous and s-open mappings. Cao, Ganester and Reilly studied the links between generalized closed sets and ex- termal disconnectedness [13]. Dontchev and Ganster [16] studied -generalized closed sets. Semi continuous and semi-closed mappings were further studied by Ghosh [21] in 1990. L. A. Zadeh [59] introduced the seminal notion of fuzzy sets in 1965. C. L. Chang [14] de...ned and studied the notion of a fuzzy topological space in 1968. Since then much attention has been paid to generalize the basic concepts of Classical Topology in fuzzy setting and thus a modern theory of Fuzzy Topology has been developed. Azad [9] fuzzi...ed the work of Levine, and presented some general properties of fuzzy spaces. Several properties of semi- open fuzzy (resp. semi-closed fuzzy ), fuzzy regular open (resp. closed) sets have also been discussed by Azad. Abbas [4] studied fuzzy super irresolute mappings, Ajmal and Azad [5, 6] gave pointwise characterization of fuzzy almost continuity. Caldas, Navalagi and Saraf [10, 11] gave a study of fuzzy weakly semi-open mappings. Jankovic [23] introduced the notion of -regular spaces. In 2002 Georgiou and Papadopoulos [20] studied fuzzy -convergences. Ming and Ming [36] de...ned the notion of fuzzy boundary in fuzzy topo- logical spaces in 1980, yet there is very little work available on this notion, in present literature. Tang [54] used a limited version of Chang’ fuzzy topolog- s ical space because su¢ cient material about properties of fuzzy boundary is currently not available. So, we study this concept and establish several of its properties in Chapter 2. We also de...ne the concept of semi fuzzy -boundary and characterized semi-continuous fuzzy functions in terms of semi fuzzy - boundary. Several properties of fuzzy boundary and semi fuzzy -boundary have been obtained, which have been supported by examples. Properties of semi fuzzy -interior, semi fuzzy -closure, fuzzy boundary and semi fuzzy - boundary have been obtained in product related spaces. We give necessary conditions for continuous fuzzy (resp. semi-continuous fuzzy, irresolute fuzzy) functions. Moreover, continuous fuzzy (resp. semi-continuous fuzzy, irresolute fuzzy) functions have been characterized via derived fuzzy(resp. semi-derived fuzzy) sets. The results of this chapter have been published in Advances in Fuzzy Systems Vol. 2008, Article ID 586893, 9 pages doi:10.1155/2008/586893 (MR# 2425456). In Chapter 3, we studied semi-continuous fuzzy, semi-open fuzzy and al- most open fuzzy (Ganguly’ sense) mappings. We also de...ne and study prop- s erties of almost closed fuzzy mappings. In Chapter 4, we continue the study initiated in Chapter 3 and several properties and characterizations of semi- open fuzzy (semi-closed fuzzy), semi-preopen fuzzy (semi-preclosed fuzzy), semi-precontinuous fuzzy and pre-semi-preopen fuzzy (pre-semi-preclosed fuzzy) mappings have been investigated. Findings of Chapters 3 and 4 have been pub- lished in Journal of Fuzzy Mathematics, 16(2)(2008), 341-349 (Zbl# 1146.54302) and vol. 18(1), respectively. In Capter 5, we further study some properties of semi-open fuzzy sets de- ...ned and studied by Zhong [62], semi-preopen fuzzy sets and preopen fuzzy sets. It is also shown that in the class of injective functions, almost open fuzzy (closed) in Nanda’ sense and almost quasi-compact fuzzy functions are equiv- s alent. In terms of graph and projections, some interesting characterizations and properties of almost continuous fuzzy functions in Singal’ sense are given. s Moreover almost continuous fuzzy in Husain’ sense, almost weakly continu- s ous fuzzy,nearly almost open (closed) fuzzy functions have been de...ned and their several characterizations and properties have been obtained. Finally, their equivalences have been established under certain conditions. Results from this chapter have appeared in International Journal of Contemporary Mathematical Sciences 3(34) (2008) 1665-1677 (MR# 2511023). In 2001, Kresteska [29] pointed out that Lemmas 4.5, 4.7 and Theorems 4.6, 4.8, 4.12 of [52] are incorrect. Since -continuity does not yield to a straightforward fuzzi...cation of the results from Classical Topology, thus this notion seems promising for Fuzzy Topology. Motivated by such consideration, Chapter 6 studies further, the properties of -continuous mappings in terms of -closure of fuzzy sets. Findings of this chapter have been submitted to Korean Annals of Mathematics. In Chapter 7, our aim is to further contribute to the study of semi-open fuzzy sets by establishing several important fundamental identities and in- equalities about their semi-interior and semiclosure. D. E. Cameron and G. Woods [12] introduced the concepts of s-continuous mappings and s-open map- pings. They investigated the properties of these mappings and their relation- ships to properties of semi-open sets. M. Khan and B. Ahmad [25] further worked on the characterizations and properties of s-continuous, s-open and s-closed mappings. In this section, we fuzzify the ...ndings of [12] and [25]. We de...ne s-open and s-closed fuzzy mappings and establish some interesting char- acterizations of these mappings. It may be noted that the class of s-open (resp. s-closed) fuzzy mappings is a subclass of the class of open (resp. closed) fuzzy mappings. These results have been published by Advances in Fuzzy Systems Volume 2009 (2009), Article ID 303042, 5 pages doi:10.1155/2009/303042. Chapter 8 comprises a study of simply continuous fuzzy mappings. In Chapter 9, we de...ne and study the notion of -semicontinuous fuzzy map- pings. Results of this capter have been submitted for publication.