Fuzzy W-closed sets

In this paper, we introduce the relatively new notion of fuzzy W-closed and fuzzy W-generalized closed sets. Several properties and connections to other well-known weak and strong fuzzy closed sets are discussed. Fuzzy W -generalized continuous and fuzzy W-generalized irresolute functions and their basic properties and relations to other fuzzy continuities are explored.

Let (X, ) be a fuzzy topological space (simply, Fts). If is a fuzzy set (simply, F-set), then the closure of and the interior of will be denoted by Cl ( ) and Int ( ), respectively. If no ambiguity appears, ≤ Int (Cl ( )). Finally, is called fuzzy generalized closed (simply, FGC) if for every FO-set with ≤ , we have Cl ( ) ≤ . The collection of all FSO (resp., FSC and FGC) subsets of X will be denoted by FSO(X, ) (resp., FSC(X, ) and FGC(X, ) ). A fuzzy space is called an E.D. space if the closoure of every FSO-set in it is FO. Equivalently, the interior of every FSC-set in it is SC. A fuzzy function f :(X, ) → (Y, � ) is called fuzzy continuous (simply, FCts) if the inverse image of every FC-set is FC.

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We introduce the relatively new notion of fuzzy weak closed set and fuzzy weak generalized closed sets. Several properties and connections to other well-known weak and strong fuzzy closed sets are discussed. Fuzzy weak generalized continuous and fuzzy weak generalized irresolute functions and their basic properties and relations to other fuzzy continuities are explored.
In this paper, we introduce the relatively new notions of fuzzy W-closed sets, which is closely related to the class of fuzzy closed subsets. We investigate several characterizations of fuzzy W-open and fuzzy W-closed notions via the operations of interior and closure, for more on these notions for crisp sets see Al-Hawary (2004, 2013a, 2013b and Al-Hawary and Al-Omari (2006, 2008, 2009). In Section 3, we introduce the notion of fuzzy W-generalized closed sets and study connections to other weak and strong forms of fuzzy generalized closed sets. Section 4 is devoted to introducing and studying fuzzy W-generalized continuous and fuzzy W-generalized irresolute functions and connections with other similar forms of fuzzy continuity.

Fuzzy W-closed sets
We begin this section by introducing the notions of fuzzy W-open and fuzzy W-closed subsets.
Definition 1 Let be a fuzzy subset of a Fts space (X, ). The fuzzy W -interior of is the union of all fuzzy open subsets of X whose closures are contained in Cl( ), and is denoted by Clearly Int ( ) ≤ Int W ( ) ≤ Cl ( ) and ≤ Cl ( ) ≤ Cl W ( ) and hence every FWC−set is a FC-set, but the converses needs not be true.
Even the intersection of two FWO subsets needs not be FWO. In classical topology, the interior of a set is a subset of the set itself. But this is not the case for FWO-sets. Next we show that ≤ Int W ( ) and Int W ( ) ≤ need not be true. Corollary 2 If is a fuzzy −dense subset of X (Cl ( ) = 1), then Int ( ) = 1.
We remark that X being an E. D. fuzzy space is necessary in Lemma 2.

Fuzzy W-generalized closed sets
In this section, we introduce the notion of fuzzy W-generalized closed set. Moreover, several interesting properties and constructions of these fuzzy subsets are discussed. Let be a FWGO-set set and be a FC subset such that ≤ . Then 1 − ≤ 1 − . As 1 − is FWGC and as 1 − is FO, Next we show the class of FWGC-sets is properly placed between the classes of FC-and FWC-sets. Moreover, the class FGC-sets is properly placed between the classes of FC-sets and FWGC−sets. A FC-set is trivially FGC and clearly every FWC-set is FC and every FWGC-set is FGC as Cl( ) ≤ Cl W ( ) for every F-set in space X. In Example 1, = {a,c} is a FC-set that is not FWC. In Example 2, = {a,b,d} is not FWC, but as the only super set of is 1, is FWGC. The following is an immediate result from Lemma 2: Theorem 3 If is a FSC subset of a fuzzy E.D. space X, the following are equivalent: (1) is a FWGC − set; (2) is a FGC − set.
Since and Finally since ∧ is a F-subset of and , Cl W ( ∧ ) ≤ Cl W ( ) ∧ Cl W ( ).

Corollary 5 Finite unions of FWGC-sets are FWGC.
While the finite intersections of FWGC-sets need not be FWGC.

Theorem 4 The intersection of a FWGC-set set with a FWC-set is FWGC.
Proof Let be a FWGC-set and be a FWC-set. Let be a FO-set such that ∧ ≤ . Then ≤ ∨ (1 − ). Since 1 − is FWO, by Corollary 3, ∨ (1 − ) is FO and since is FWGC,
Lemma 4 Let f :(X, ) → (Y, � ) be F Gcts. Then f is FGcts. Follows from the fact that every FWGC-set is FGC. The converse of the preceding Lemma needs not be true.
Even the composition of FWGCts functions needs not be FWGCts.
Example 8 Let f be the function in Example 6. Let ε = {0, 1, {a,b,d,e} }. Let g:(X, � ) → (X, ε ) be the identity function. It is easily observed that g is also F GCts as the only super set of {c} is 1. But the composition function g•f is not F GCts as {c} is a FC-set in (X, ε ) and it is not a F GC-set in (X, ).
We end this section by giving a necessary condition for a FWGCts function to be FWGI.