Generalized ψρ-Operations on Fuzzy Topological Spaces

and Applied Analysis 3 Definition 1.4 see 18, 20–23 . 1 The intersection of all fuzzy α-closed resp., semiclosed, preclosed, γ-closed, semi-preclosed, β-closed sets containing a fuzzy set A is called a fuzzy α-closure resp., semiclosure, preclosure, γ-closure, semi-preclosure, β-closure of A. 2 The union of all fuzzy α-open resp., semiopen, preopen, γ-open, semi-preopen, β-open sets contained in a fuzzy set A is called a fuzzy α-interior resp., semi-interior, preinterior, γ-interior, semi-preinterior, β-interior of A. Definition 1.5. A fuzzy point xr in an fts X is said to be a fuzzy cluster resp., θ-cluster 24 , δ-cluster 16 point of a fuzzy set A if and only if for every fuzzy open resp., open, regular open q-neighborhoodU of xr ,UqA resp., Cl U qA,UqA . The set of all fuzzy cluster resp., fuzzy θ-cluster, fuzzy δ-cluster points of A is called the fuzzy closure resp., θ-closure, δclosure ofA and is denoted by Cl A resp., θ-cl A , δ-cl A . A fuzzy setA is fuzzy θ-closed resp., δ-closed if and only if A θ-cl A resp., A δ-cl A . The complement of a fuzzy θ-closed resp., δ-closed set is called fuzzy θ-open resp., δ-open . Definition 1.6. A fuzzy set A of an fts X, τ is said to be: 1 fuzzy generalized closed 3 briefly, g-closed if Cl A ≤ U, whenever A ≤ U and U is a fuzzy open set in X, 2 fuzzy generalized α-closed 10 briefly, gα-closed if α-cl A ≤ U, wheneverA ≤ U and U is a fuzzy open set in X, 3 fuzzy α-generalized closed 13 briefly, αg-closed if α-cl A ≤ U, wheneverA ≤ U andU is a fuzzy α-open set in X, 4 fuzzy generalized semiclosed 12 briefly, gs-closed if Scl A ≤ U, whenever A ≤ U and U is a fuzzy open set in X, 5 fuzzy semigeneralized closed 5 briefly, sg-closed if Scl A ≤ U, wheneverA ≤ U andU is a fuzzy semiopen set in X, 6 fuzzy generalized preclosed 8 briefly, gp-closed if Pcl A ≤ U, wheneverA ≤ U and U is a fuzzy open set in X, 7 fuzzy pregeneralized closed 6 briefly, pg-closed if Pcl A ≤ U, wheneverA ≤ U andU is a fuzzy preopen set in X, 8 fuzzy generalized semi-preclosed 11 briefly, gsp-closed if Spcl A ≤ U, whenever A ≤ U and U is a fuzzy open set in X, 9 fuzzy semi-pregeneralized closed 14 briefly, spg-closed if Spcl A ≤ U, whenever A ≤ U and U is a fuzzy semi-preopen set in X, 4 Abstract and Applied Analysis 10 fuzzy regular generalized closed 9 briefly, rg-closed if Cl A ≤ U, whenever A ≤ U and U is a fuzzy regular open set in X, 11 fuzzy generalized θ-closed 4 briefly, gθ-closed if θ-cl A ≤ U, whenever A ≤ U andU is a fuzzy open set in X, 12 fuzzy θ-generalized closed 7 briefly, θg-closed if θ-cl A ≤ U, whenever A ≤ U andU is a fuzzy θ-open set in X, 13 fuzzy δθ-generalized closed 15 briefly, δθg-closed if δ-cl A ≤ U, whenever A ≤ U and U is a fuzzy θ-open set in X. The complement of a fuzzy generalized closed resp., generalized α-closed, αgeneralized closed, generalized semiclosed, semigeneralized closed, generalized preclosed, pre generalized closed, generalized semi-preclosed, semi-pregeneralized closed, regular generalized closed, θ-generalized closed, generalized θ-closed set is called fuzzy generalized open g-open, for short resp., generalized α-open gα-open , α-generalized open αg-open , generalized semiopen gs-open , semi generalized open sg-open , generalized preopen gp-open , pre generalized open pg-open , generalized semi-preopen gsp-open , semipregeneralized open spg-open , regular generalized open rg-open , θ-generalized open θg-open , generalized θ-open gθ-open . Definition 1.7 see 25 . A fuzzy point xr in an fts X, τ is called weak resp., strong if r ≤ 1/2 resp., r > 1/2 . 2. ψ-Operations In this research, we will denote for a fuzzy open set from type ψ by fuzzy ψ-open and the family of all fuzzy ψ-open sets in an fts X, τ by ψO X . Also we will denote a fuzzy open resp., α-open, semiopen, preopen, semi-preopen, γ-open, β-open, δ-open, θ-open, and regular open set by τ-open resp., α-open, s-open, p-open, sp-open, γ-open, β-open, δ-open, θ-open, and r-open . Similarly we will denote a fuzzy closed resp., α-closed, semiclosed, preclosed, semi-preclosed, γ-closed, β-closed, δ-closed, θ-closed, and regular closed sets by τ-closed resp., α-closed, s-closed, p-closed, sp-closed, γ-closed, β-closed, δ-closed, θ-closed, and r-closed . Let Ω {τ, α, s,p, sp, γ, β, δ, θ, r}. Definition 2.1. A fuzzy set A in an fts X, τ is said to be a fuzzy ψ-q-neighborhood of a fuzzy point xr if and only if there exists a fuzzy ψ-open setU such that xrqU ≤ A. The family of all fuzzy ψ − q-neighborhoods of a fuzzy point xr is denoted by N ψ xr . Definition 2.2. A fuzzy point xr in an fts X, τ is said to be a fuzzy ψ-cluster point of a fuzzy setA if and only if for every fuzzy ψ − q-neighborhoodU of a fuzzy point xr ,UqA. The set of all fuzzy ψ-cluster points of a fuzzy setA is called the fuzzy ψ-closure ofA and is denoted by ψcl A . A fuzzy set A is fuzzy ψ-closed if and only if A ψcl A and a fuzzy set A is fuzzy ψ-open if and only if its complement is fuzzy ψ-closed. Theorem 2.3. For a fuzzy set A in an fts X, τ , ψcl A ∧F : F ≥ A, 1 − F ∈ ψO X . 1 Abstract and Applied Analysis 5and Applied Analysis 5 Proof. The proof of this theorem is straightforward, so we omit it. Theorem 2.4. Let A and B be fuzzy sets in an fts X, τ . Then the following statements are true: 1 ψcl 0 0, ψcl 1 1; 2 A ≤ ψcl A for each fuzzy set A of X; 3 if A ≤ B, then ψcl A ≤ ψcl B ; 4 ifA is ψ-closed, thenA ψcl A , and if one supposes ψcl A is ψ-closed, then the converse of (4) is true; 5 if V ∈ ψO X , then VqA if and only if Vqψcl A ; 6 ψcl ψcl A ψcl A ; 7 ψcl A ∨ψcl A ≤ ψcl A∨B . If the intersection of two fuzzy ψ-open sets is fuzzy ψ-open, then ψcl A ∨ ψcl A ψcl A ∨ B . Proof. 1 , 2 , 3 , and 4 are easily proved. 5 Let VqA. Then A ≤ 1 − V , and hence ψcl A ≤ ψcl 1 − V 1 − V , which implies Vqψcl A . Hence VqA if and only if Vqψcl A . 6 Let xr be a fuzzy point with xr / ∈ ψcl A . Then there is a fuzzy ψ − q-neighborhood U of xr such that UqA. From 5 there is a fuzzy ψ − q-neighborhood U of xr such that Uqψcl A and hence xr / ∈ ψcl ψcl A . Thus ψcl ψcl A ≤ ψcl A . But ψcl ψcl A ≥ ψcl A . Therefore ψcl ψcl A ψcl A . 7 It is clear. Definition 2.5. For a fuzzy set A in an fts X, τ , we define a fuzzy ψ-interior of A as follows: ψint A ∨U : U ≤ A,U ∈ ψO X . 2 Theorem 2.6. Let A and B be fuzzy sets in an fts X, τ . Then the following statements are true: 1 ψint 0 0, ψint 1 1; 2 A ≥ ψint A for each fuzzy set A of X; 3 if A ≤ B, then ψint A ≤ ψint B ; 4 ifA is ψ-open, thenA ψint A , if one supposes, ψint A is ψ-open, then the converse of (4) is true; 5 if V ∈ ψC X , then VqA if and only if Vqψint A ; 6 ψint ψint A ψint A ; 7 ψint A ∧ ψint A ≥ ψint A ∧ B . If the intersection of two fuzzy ψ-open sets is ψ-open, then ψint A ∧ ψint A ψint A ∧ B . Proof. It is similar to that of Theorem 2.4. 6 Abstract and Applied Analysis Theorem 2.7. For a fuzzy set A in an fts X, τ , the following statements are true: 1 ψcl 1 −A 1 − ψint A ; 2 ψint 1 −A 1 − ψcl A . Proof. It follows from the fact that the complement of a fuzzy ψ-open set is fuzzy ψ-closed and ∨ 1 −Ai 1 − ∧Ai. Definition 2.8. Let A be a fuzzy set of an fts X, τ . A fuzzy point xr is said to be ψ-boundary of a fuzzy set A if and only if xr ∈ ψcl A ∧ 1 − ψcl A . By ψ-Bd A one denotes the fuzzy set of all ψ-boundary points of A. Theorem 2.9. Let A be a fuzzy set of an fts X, τ . Then A ∨ ψ-Bd A ≤ ψcl A . 3 Proof. It follows from Definition 2.8 and Theorem 2.4. 3. Generalized ψρ-Closed and Generalized ψρ-Open Sets Definition 3.1. Let X, τ be an fts. We define the concepts of fuzzy generalized ψρ-closed and fuzzy generalized ψρ-open sets, where ψ represents a fuzzy closure operation and ρ represents a notion of fuzzy openness as follows: 1 A fuzzy setA is said to be generalized ψρ-closed gψρ-closed, for short if and only if ψcl A ≤ U, whenever A ≤ U and U is fuzzy ρ-open. 2 The complement of a fuzzy generalized ψρ-closed set is said to be fuzzy generalized ψρ-open gψρ-open, for short . Remark 3.2. Note that each type of generalized closed set in Definition 2.8 is defined to be generalized ψρ-closed set for some ψ ∈ Ω \ {r} and ρ ∈ Ω. Namely, a fuzzy set A is fuzzy g-closed 3 if it is gττ-closed, gα-closed 10 if it is gατ-closed, αg-closed 13 if it is gααclosed, gs-closed 12 if it is gsτ-closed, sg-closed 5 if it is gs − s-closed, gp-closed 8 if it is gpτ-closed, pg-closed 6 if it is gp − p-closed, gsp-closed 11 if it is gspτ-closed, spg-closed 14 if it is gsp − sp-closed, gθ-closed 4 if it is gθτ-closed, θg-closed 7 if it is gθθ-closed, and gr-closed 9 if it is gτr-closed. Theorem 3.3. A fuzzy set A is generalized ψρ-open if and only if ψint A ≥ F, whenever A ≥ F and F is fuzzy ρ-closed.