Generalized Fuzzy Closed Sets in Smooth Bitopological Spaces [chapter]

Osama A. El-Tantawy, Sobhy A. El-Sheikh, Rasha N. Majeed
Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing  
In [34] we introduced the notion of r-(τ i , τ j )-θ-generalized fuzzy closed sets in smooth bitopological spaces by using (τ i , τ j )θ-fuzzy closure T τi τj defined in [19] . Recently, [33] we defined a new θ-fuzzy closure, denoted C θ 12 on smooth bitopological spaces by using smooth supra topological space (X, τ 12 ) which is generated from smooth bitopological space (X, τ 1 , τ 2 ) [1], such that C θ 12 ≤ T τi τj . In this paper, we introduce a new class of r-θ-generalized fuzzy closed
more » ... ed fuzzy closed sets, namely, r-τ 12 -θ-gfc in smooth bitopological spaces via C θ 12 -fuzzy closure operator. The basic properties of these sets are studied. Furthermore, the relationship with other notions of r-generalized fuzzy closed sets in [31, 32, 33, 34] are investigated and we give many examples for reverse. In addition, by using r-τ 12 -θ-gfc sets, we define a new fuzzy closure operator which generates a new smooth topology. Finally, generalized fuzzy θ-continuous (resp. irresolute) and fuzzy strongly θ-continuous mappings are introduced and some of their properties are studied. 61 The so-called supra topology was established, by Mashhour et al. [24] (recall that a supra topology on a set X is a collection of subsets of X, which is closed under arbitrary unions). Abd El-Monsef and Ramadan [2] introduced the concept supra fuzzy topology, followed by Ghanim et al. [13] who introduced the supra fuzzy topology inŠostak sense. Abbas [1] generated the supra fuzzy topology (X, τ 12 ) from fuzzy bitopological space (X, τ 1 , τ 2 ) inŠostak sense as an extension of supra fuzzy topology due to Kandil et al. [15]. The first attempt of generalizing closed sets was done by Levine [23]. Subsequently, Fukutake [12], generalized this concept in bitopological space. Balasubramanian and Sundaram [4], introduced the concept of generalized fuzzy closed sets within Chang's fuzzy topology. Kim and Ko [18] defined r-generalized fuzzy closed sets in smooth topological spaces. Recently, in [31], we introduced the concept of generalized fuzzy closed sets in smooth bitopological spaces. Noiri [25] and Dontchev and Maki [8] introduced another new generalization of Livine generalized closed set by utilizing the θ-closure operator. The concept of θ-generalized closed sets was applied to the digital line [9]. Khedr and Al-Saadi [16] generalized the notion of θ-generalized sets to bitopological space. El-Shafei and Zakari [10] introduced the concept of θ-generalized fuzzy closed sets in Chang's fuzzy topology. Recently, in [34] , we introduced the notion of θ-generalized fuzzy closed sets in smooth bitopological spaces by utilizing the (τ i , τ j )θ-fuzzy closure T τi τ j defined in [19] . In this paper we define another type of r-θ-generalized fuzzy closed sets in smooth bitopological spaces via C θ 12 -fuzzy closure which was established by us [33] , and study its relationship with other types of r-generalized fuzzy closed sets which introduced in ([31, 32, 33, 34]). By using this new class of generalized fuzzy closed sets we define a new fuzzy closure operator which generates a new smooth topology. Finally, we define and study generalized fuzzy θ-continuous (resp. irresolute) and fuzzy strongly θ-continuous mappings.
doi:10.4018/978-1-4666-9798-0.ch020 fatcat:qpahwpzr4fbvph3drwyhmnp7ai