Inventi Impact: Fuzzy System Vol. 2012, Issue 3 [ISSN 2277-2316]

2012 efs 025, CCC: $10 © Inventi Journals (P) Ltd Published on Web 15/07/2012, www.inventi.in

RESEARCH ARTICLE

1. INTRODUCTION In [4], Othman has introduced the concept of fuzzy sp-open sets. Several notions based on fuzzy sp-open (closed) set and fuzzy sp-continuous mapping have been studied. Moreover, the study also included the relationships between those concepts and some other weaker forms of fuzzy open sets and fuzzy continuous mappings. In [1], Azad introduced the concept of fuzzy almost continuity and fuzzy weakly continuity. The purpose of this paper is to introduce the notions of fuzzy almost sp-continuous mapping, fuzzy weakly sp-continuous mapping and fuzzy faintly sp-continuous mapping and to investigate some properties of these notions and the relationship between them. Furthermore using sp-connectedness and the new concept sp-compactness we prove some properties of weakly sp-continuous mappings. 2. PRELIMINARIES Throughout this paper by (X, τ) or simply by X we mean a fuzzy topological space (fts, shortly) and f : X → Y means a mapping f from a fuzzy topological space X to a fuzzy topological space Y . If u is a fuzzy set and p is a fuzzy singleton in X, then N (p), Int u, cl u, uc, clδ u, sp – Int u, sp – cl u, FSP − O(X) and FSP − C(X), will denote respectively, the neighbourhood system of p, the interior of u, the closure of u, complement of u, the fuzzy δ-closure of a fuzzy set u, the fuzzy sp-interior of a fuzzy set u, the fuzzy sp-closure of a fuzzy set u, the family of all fuzzy sp-open sets of (X, τ ) and the family of all fuzzy sp-closed sets of (X, τ ).

Now, we mention the following definitions and results which are used in this paper concerning fuzzy topology. Definition 2.1 [3] A fuzzy singleton p in X is a fuzzy set defined by: p(x) = t, for x = x0 and p(x) = 0 otherwise, where 0 < t ≤ 1. The point p is said to have support x0 and value t. Definition 2.2 A fuzzy set λ in a f ts X is called fuzzy preopen [5], (resp. fuzzy sp-open [4], fuzzy δ-open) set if λ ≤ Int cl λ (resp. λ ≤ Int cl (λ) ∨ cl Int (λ), for x ∈ λ, there exists a regular open set µ such that x ∈ µ ≤ λ. The family of all fuzzy preopen (resp. fuzzy sp-open) sets of X is denoted by F P O(X) (resp. FSP − O(X), FδO (X)).

1Department of Mathematics, University College of Alqunfudah, Umm Alqura University, Mecca, Saudi Arabia. 2Department of Mathematics, College of Education and Science, Rada'a, Albayda University, Albayda, Republic of Yemen. E-mail: hakimalbdoie@yahoo.com *Correspondig author

Theorem 2.3 [4] For a fuzzy subset λ of a fuzzy space X, the following statements hold: sp − cl λ ≥ λ ∨ (Int cl λ ∧ cl Int λ). sp − Int λ ≤ λ ∧ (Int cl λ ∨ cl Int λ). Definition 2.4 A mapping f: (X, τ) → (Y, σ) is said to be: Fuzzy almost continuous [1] if f −1 (µ) is fuzzy open set

in X for each fuzzy regular open set µ in Y. Fuzzy weakly continuous [1] if for each fuzzy open set µ

of Y, f−1 (µ) ≤ Int f −1 (cl µ). Fuzzy sp -continuous [4] if f −1(µ) is fuzzy sp-open (fuzzy

sp-closed) set in X for each fuzzy open (fuzzy closed) set µ in Y

3. MAIN RESULTS On this section fuzzy almost sp-continuous, fuzzy weakly sp-continuous, fuzzy faintly sp-continuous and fuzzy almost open mappings and fuzzy sp-compactness are defined and some interesting properties related to these definitions are obtained. Definition 3.1 A mapping f : (X, τ ) → (Y, σ) is said to be fuzzy weakly sp- continuous if for each fuzzy singleton p ∈ X and each fuzzy open set µ of Y containing f (p), there exists λ ⊆ F SP − O(X) containing p such that f (λ) ≤ cl (µ). Theorem 3.2 For a mapping f: (X, τ) → (Y, σ), the following statements are equivalent: f is a fuzzy weakly sp-continuous at fuzzy singleton p ∈ X; A fuzzy singleton p ∈ cl Int (f −1(cl(µ))) ∨ Int cl (f−1(cl µ))

for each fuzzy neighbourhood µ of f (p); f −1(µ) ≤ sp − Int (f −1(clµ)) for every fuzzy open set µ of Y ; f −1(µ) ≤ cl Int (f −1(cl(µ))) ∨ Int (cl (f−1(cl µ))) for every

fuzzy open set µ of Y . Proof: (i) ⇒ (ii). Let µ be any fuzzy neighbourhood of f (p). Since f is weakly sp-continuous at all fuzzy singleton p ∈ X, there exists λ ⊆ F SP−O(X) such that f (λ) ≤ cl (µ). Then λ ≤ f

−1 (clµ). Since λ is fuzzy sp-open such that p ∈ λ ≤ cl Int (λ) ∨ Int cl (λ) ≤ cl Int (f −1 (cl µ)) ∨ Int cl (f −1 (clµ)). (ii) ⇒ (iii). Let a fuzzy singleton p ∈ f −1 (µ), so f (p) ∈ µ. Where µ is open. Then p ∈ f −1(cl(µ)) and since p ∈ cl Int (f

−1(cl (µ))) ∨ Int cl (f−1(clµ)) we have p ∈ f −1(clµ) ∧ [cl Int (f

−1(cl(µ))) ∨ Intcl (f−1(clµ))] ≤ sp −Int(f −1(cl(µ)). Hence f−1 (µ) ≤ sp − Int (f−1 (cl (µ)).

Some Weaker Forms of Fuzzy SP-continuous Mappings

Hakeem A Othman1,2*

Abstracts: The aim of this paper is to introduce some new weaker forms of fuzzy sp continuity, namely fuzzy almost sp-continuous mappings; fuzzy weakly sp-continuous mappings and fuzzy faintly sp-continuous mappings by using the notion of fuzzy sp-open sets. Certain fundamental properties, some new results related to these new concepts are obtained, fuzzy sp-compact is introduced and the relations and inverse relations between these new fuzzy mappings are investigated.

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Inventi Impact: Fuzzy System Vol. 2012, Issue 3 [ISSN 2277-2316]

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RESEARCH ARTICLE

(iii) ⇒ (IV). Let µ be any fuzzy open set of Y. By Theorem (2. 3) we have f −1 (µ) ≤ sp − Int (f −1 (clµ)) ≤ cl Int (f −1 (cl (µ))) ∨ Int (cl (f−1 (clµ))). (iv) ⇒ (ii). Let p be any fuzzy singleton of X and µ any fuzzy open set of Y containing f(x).Then p ∈ cl Int (f −1(cl (µ))) ∨ Int cl (f −1(cl µ)). (iii) ⇒ (i). Let µ be any neighbourhood of f (x) then p ∈ f −1 (µ) ≤ sp − Int (f−1 (cl (µ)). Put λ = sp − Int (f −1 (cl (µ)) then λ ⊆ F SP − O(X) and f (λ) ≤ cl (µ). This shows that f is weakly sp-continuous mapping at all fuzzy singletons p ∈ X. Definition 3.3 A mapping f: (X, τ ) → (Y, σ) is said to be fuzzy almost sp-continuous if for each fuzzy singleton p ∈ X and each fuzzy open set µ of Y containing f (p), there exists λ ⊆ F SP − O(X) containing p such that f (λ) ≤ Int cl (µ). Remark 3.4 Every fuzzy weakly continuous is fuzzy weakly sp - continuous. Fuzzy weakly continuous is implied by fuzzy sp-continuous and fuzzy almost sp-continuous is implied by fuzzy sp-continuous and implies weakly sp-continuous.

By using the same technique as in the proof of Theorem (3. 2), Definition (3. 3) and Remark (3. 4), we can prove the following theorems. Theorem 3.5 For a mapping f: (X, τ) → (Y, σ), the following statements are equivalent: f is a fuzzy almost sp-continuous at fuzzy singleton p ∈ X; A fuzzy singleton p ∈ cl Int (f −1(scl (µ))) ∨ Int cl (f −1(scl

µ)) for each fuzzy neighbourhood µ of f (p); f −1(µ) ≤ sp − Int (f −1(scl µ)) for every fuzzy open set µ of Y ; f −1(µ) ≤ cl Int (f −1(scl(µ))) ∨ Int (cl (f−1(scl µ))) for every

fuzzy open set µ of Y . Theorem 3.6 For a mapping f: (X, τ) → (Y, σ), the following statements are equivalent: f is a fuzzy almost sp-continuous; f −1(µ) ⊆ SP − O(X) for every fuzzy regular open µ of Y; f −1(u) ⊆ SP − C(X) for every fuzzy regular closed u of Y; f (sp – cl (λ)) ≤ clδ f (λ) for every subset λ of X; f−1 (u) ⊆ SP − C(X) for every fuzzy δ-closed u of Y; f −1(µ) ⊆ SP − O(X) for every fuzzy δ-open µ of Y . Theorem 3.7 For a mapping f: (X, τ) → (Y, σ), the following statements are equivalent: f : (X, τ ) → (Y, σ) is a fuzzy almost sp-continuous; f : (X, τ ) → (Y, σs) is a fuzzy sp-continuous; f : (X, τ sp) → (Y, σ) is a fuzzy almost continuous; f: (X, τ sp) → (Y, σs) is a fuzzy continuous. where τ sp and σs denote the family of all fuzzy sp-open sets of (X, τ) and family of all regular open sets of (Y, σ) respectively. Definition 3.8 A mapping f: (X, τ) → (Y, σ) is said to be fuzzy almost sp- open if f (λ) ≤ Intcl (f (λ)) for every fuzzy sp-open set λ of X.

Theorem 3.9 If f: (X, τ) → (Y, σ) is fuzzy almost sp-open and fuzzy weakly sp-continuous mapping, then f is almost sp-continuous mapping. Proof: Let fuzzy singleton p belong to X and let µ be a fuzzy open set of Y containing f (p). Since f is a fuzzy weakly sp-continuous mapping, there exists λ ⊆ SP −O(X) such that f (λ) ≤ cl (µ). Since f is a fuzzy almost sp-open mapping, f (λ) ≤ Int cl (f (λ)) ≤ Int cl λ and hence f is an almost sp-continuous mapping. Theorem 3.10 If f :( X, τ) → (Y, σ) is fuzzy weakly sp-continuous and g: (Y, σ) → (Z, θ) is fuzzy continuous, then the composition g o f: (X, τ) → (Z, θ) is fuzzy weakly sp-continuous. Proof: Let µ be a fuzzy open set of Z containing g (f (x)). Then g−1 (µ) is a fuzzy open set of Y containing f (x) and there exists λ ⊆ SP − O(X) such that F (λ) ≤ cl (g−1 (µ)), since g is fuzzy continuous. We obtain (g of) (λ) ≤ g (cl (g−1 (µ))) ≤ cl (µ). Definition 3.11 A mapping f : (X, τ ) → (Y, σ) is said to be fuzzy faintly sp-continuous if for each fuzzy singleton p ∈ X and each fuzzy θ-open set µ of Y containing f (p), there exists λ ⊆ F SP − O(X) containing p such that f (λ) ≤ cl (µ).

By using the definitions (2.4), (3.1), (3.3) and (3.11) we can prove this theorem. Theorem 3.12 The implication (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) holds for the following properties of a fuzzy mapping f: X → Y. f is sp-continuous; f −1( clδ (µ)) is fuzzy sp-closed in X for every fuzzy subset

µ of Y ; f is almost sp-continuous; f is weakly sp-continuous; f is faintly sp-continuous; Definition 3.13 [2] A fuzzy topological space (X, τ) is almost compact iff every open cover of X has a finite sub collection whose closures cover X.

Now, the new concept fuzzy sp-compact space is introduced. Definition 3.14 A fuzzy topological space (X, τ) is sp-compact iff every sp-open cover of X has a finite subcover. Theorem 3.15 If f: X→ Y is a fuzzy weakly sp-continuous surjective mapping and X is a fuzzy sp-compact, then Y is a fuzzy almost compact space. Proof: Let {µi: i ∈ J} be a cover of Y by fuzzy open sets of Y . For each fuzzy singleton x ∈ X, there exists i(x) ∈ J such that

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f (x) ∈ µi(x). Since f is fuzzy weakly sp-continuous, there exists a fuzzy sp-open set λx of X containing x such that f (λx) ≤ cl (µi(x)). The family {λx: x ∈ X} is a fuzzy sp-open cover of X by fuzzy sp-open sets of X and hence there exists a finite subset Xo of X such that X ≤ ∨x∈Xo λx. Therefore, we obtain Y = f(X) ≤ ∨x∈Xo cl (µi(x)). This show that Y is fuzzy almost compact. Definition 3.16 [4] A fuzzy set v in a fuzzy topological space (X, τ) is said to be fuzzy sp-connected if and only if µ cannot be expressed as the Union of two fuzzy sp-separated sets.

By using the definition of fuzzy sp-connectedness, we obtain the following. Theorem 3.17 Let f: X → Y be a fuzzy weakly sp- continuous surjective mapping and X is fuzzy sp-connected then Y is fuzzy connected. Proof: Suppose that Y is not connected. There exists nonempty fuzzy open sets µ and λ of Y such that µ ∨ λ = Y and µ ∧ λ = 0x. Then µ and λ are clopen in Y. By Theorem (3. 2) we have f −1 (µ) ≤ sp− Int (f −1 (clµ)) = sp − Int (f −1 (µ)) and hence f −1 (µ) is fuzzy sp-open in X. Similarly f −1 (λ) is fuzzy sp-open in X. Moreover we have f−1 (µ) ∧ f−1 (λ) =0x, f−1 (µ) ∨ f−1 (λ =X and f−1 (µ) and f−1 (λ) are nonempty. Therefore X is not fuzzy sp-connected. Theorem 3.18 If f: X → Y is a fuzzy faintly sp-continuous surjective mapping and X is a fuzzy sp-compact, then Y is a fuzzy θ compact space. Proof: Let {µj: j ∈ I} be the fuzzy θ-open cover of Y . Since f is fuzzy faintly sp-continuous. {f −1{µj : j ∈ I}} is a collection of fuzzy sp-open sets in X. Since X is a fuzzy sp-compact, there exists a finite subset Io of I such that X ⊆ {∨j f−1 (µj): j ∈Io}. Then Y =f(X) ⊆ {µj: j ∈Io} this shows that Y is fuzzy θ compact.

In the next three theorems, we use the condition of fuzzy regular space to achieve the inverse relations of these mappings.

It is easy to prove that fuzzy sp-continuous mapping implies fuzzy almost sp-continuous mapping. Theorem 3.19 If Y is a fuzzy regular space, then a mapping f: X → Y is a fuzzy sp-continuous if and only if f is a fuzzy almost sp-continuous. Proof: Since Y is a fuzzy regular space, there exists a fuzzy open set G of f (p), such that f (p) ∈ cl G ≤ µ. Since f is a fuzzy

almost sp-continuous, for every fuzzy singleton p ∈ X and every µ ∈ N (f (p)), there exists a fuzzy sp-open λ, such that p ∈ λ and f (λ) ≤ Int cl G. Since clG ≤ µ and int cl G ≤ Intµ (µ is fuzzy open set). Therefore f (λ) ≤ µ, hence f is a fuzzy sp-continuous.

It is easy to prove that a fuzzy sp-continuous mapping implies a fuzzy weakly sp-continuous mapping. Theorem 3.20 If Y a fuzzy regular space, then a mapping f: X → Y is a fuzzy sp-continuous if and only if f is a fuzzy weakly sp-continuous. Proof: Since Y is a fuzzy regular space, there exists a fuzzy open set G of f (p), such that f (p) ∈ cl G ≤ µ. Since f is a fuzzy weakly sp-continuous, then for every fuzzy singleton p ∈ X and every µ ∈ N (f (p)), there exists a fuzzy sp-open λ, such that p ∈ λ and f (λ) ≤ cl G. Since cl G ≤ µ, f (λ) ≤ µ, hence f is a fuzzy sp-continuous. Theorem 3.21 If Y a fuzzy regular space, then a mapping f: X → Y is a fuzzy almost sp-continuous if and only if f is a fuzzy weakly sp continuous. Proof: The proof follows from Theorem (3. 19) and Theorem (3. 20). Corollary 3.22 If Y a fuzzy regular space, the following is equivalent for a mapping f: X → Y. f is fuzzy sp-continuous; f is fuzzy almost sp-continuous; f is fuzzy weakly sp-continuous.. REFERENCES AND NOTES 1. Azad, K. K., on fuzzy semi continuity, fuzzy almost continuity

and weakly continuity, J. Math. Anal. Appl., 82 (1981) 14-32. 2. Di Concilio, A. and Gerla, G. Almost compact in fuzzy topological

spaces, Fuzzy Sets and Systems, 13(1984) 187- 192. 3. Ghanim, M. H. Kerre E. E. and Mashhour, A. S. Separation

Axioms, Sub- space and Sums in Fuzzy Topology, J. Math. Anal Appl, 102(1984) 189-202.

4. Othman, Hakeem A. On fuzzy sp-open sets, Hindawi Publishing Corporation, Advances in Fuzzy Systems Volume 2011, Article ID 768028, 5 pages, doi:10.1155/2011/768028 .

5. Singal, M. and Prakash, K. N. Fuzzy preopen sets and fuzzy pre separation axioms, Bull. Call. Math. Soc., 78 (1986) 57 - 69.

Cite this article as: Hakeem A Othman. Some Weaker Forms of Fuzzy SP-continuous Mappings. Inventi Impact: Fuzzy System, 2012(3):158-160, 2012.

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