/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 2, 2015, 185-200 ISSN 1307-5543 – www.ejpam.com New Type of Strongly Continuous Functions In topological Spaces Via δ− β-Open Sets Alaa Mahmood Farhan1,2, Xiao-Song Yang1 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Hongshan Area, Wuhan city, Hubei province, China 2 Department of Mathematics, Anbar University, College of Education for Pure Sciences, Al-Ramadi city, Iraq, P.O.Box: (55 Ramadi) Abstract. In this paper we introduce and investigate a new class of strong continuous functions called strongly θ − δ − β -continuous functions by using two new strong forms of δ − β -open sets called δ− β -regular sets and δ− βθ -open sets. This class is a generalization of both strongly θ -e-continuous functions and strongly θ − β -continuous functions. Several new characterizations and fundamental properties concerning strongly θ −δ−β -continuous functions are obtained. Furthermore, the relation- ships between strongly θ −δ−β -continuous functions and other well-known types of strong continuity are also discussed. 2010 Mathematics Subject Classifications: 54C05,54C08,54C10 Key Words and Phrases: δ − β -open sets, δ − βθ -closed sets, δ − β -regular sets, strongly θ − δ − β - continuity 1. Introduction The notion of continuity is an important concept in general topology as well as all branches of mathematics and quantum physics of course its strong forms are important, too. Recently, strong continuity of functions in topological spaces has been introduced and investigated by many mathematicians and quantum physicists. Furthermore, generalized open and closed sets are, as well-known, the most important notions in both pure and applied mathematics.The notion of continuity by involving these notions is the subject-matter of topology which has penetrated in the whole body of science. In the course of time, mathematicians realized that it is very useful to generalize the notions of open and closed sets and accordingly the notion of continuity. In 1980, Noiri [37] introduced the notion of strong θ -continuity which is stronger than δ-continuity [37]. Some properties of strongly θ -continuous functions are studied by Long and Herrington [32]. Recently, five generalizations of strong θ -continuity are obtained Email addresses: alaa_mf1970@yahoo.com (A. Jumaili), yangxs@cqupt.edu.cn (X. Yang) http://www.ejpam.com 185 c© 2015 EJPAM All rights reserved. A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 186 by Jafari and Noiri [29], Noiri [38], Noiri and Popa [39], Park [42] and Murad ÄOzkoc and GÄulhan Aslim [41]. The purpose of the present paper is to introduce and investigate a new class of strong continuous functions called strongly θ − δ − β -continuous functions and give several characterizations and fundamental properties concerning strongly θ−δ−β -continuous functions by using δ − β -open sets due to by Erdal Ekici [9] and E. Hatir and T. Noiri [27]. Also we discussed the relationships between strongly θ−δ−β -continuous functions and other well-known types of strong continuity. 2. Preliminaries Throughout this paper, (X , T ) and (Y, T ∗) (or simply X and Y ) mean topological spaces on which no separation axioms are assumed unless explicitly stated. For any subset A of X , The closure and interior of A are denoted by Cl(A) and Int(A), respectively. We recall the following definitions, which will be used often throughout this paper. A subset A of a space (X , T ) is called δ-open [48] if for each x ∈ A there exists a regular open set V such that x ∈ V ⊂ A. The δ-interior of A is the union of all regular open sets contained in A and is denoted by Intδ(A). The subset A is called δ-open [48] if A= Intδ(A). A point x ∈ X is called a δ-cluster points of A [48] if A ⋂ Int(Cl(V )) 6= φ for each open set V containing x. The set of all δ-cluster points of A is called the δ-closure of A and is denoted by Clδ(A).I f A= Clδ(A)), then A is said to be δ-closed [48]. The complement of δ-closed set is said to be δ-open set. A subset A of a space X is called δ−β -open [27] or e∗-open [9], if A⊂ Cl(Int(δ−Cl(A))), the complement of a δ−β -open set is called δ−β -closed. The intersection of all δ−β -closed sets containing A is called the δ−β -closure of A [27] and is denoted by δ−β -Cl(A). The union of all δ−β -open sets of X contained in A is called the δ−β -interior [27] of A and is denoted by δ−β -Int(A). The family of all δ−β -open (resp. δ−β -closed) subsets of X containing a point x ∈ X is denoted by δ − βΣ(X , x) (resp. δ − βC(X, x)), the family of all δ − β -open (resp. δ − β -closed) sets in X are denoted by δ − βΣ(X , T ) (resp. δ − βC(X, T)). A subset A is said to be regular open (resp. regular closed) if A = Int(Cl(A)) (resp. A=Cl(Int(A))). A subset A of X is called semiopen [31] (resp. α-open [36], preopen [33], b-open [2], semi- preopen [1] (or β -open [11]), e-open [8]) if A⊂ Cl(Int(A)) (resp. A⊂ Int(Cl(Int(A))), A⊂ Int(Cl(A)), A⊂ Int(Cl(A)) ⋃ Cl(Int(A)), A⊂ Cl(Int(Cl(A))), A ⊂ Cl(δ − Int(A)) ⋃ Int(δ − Cl(A)), and the complement of a semiopen (resp. α-open, preopen, b-open, semi-preopen, e-open) set are called semiclosed (resp. α-closed, preclosed, b-closed, semi-preclosed, e-closed ). Remark 1. Since the notion of δ− β -open sets and the notion of e∗-open sets are same, we will use the term δ− β -open sets instead of e∗-open sets. Remark 2. Erdal Ekici [8] shows that the notions of e-open set and b-open set and the notions of e-open set and β -open set and the notions of e-open set and semiopen set are independent, see example (2.6) [8]. Lemma 1 ([9, 28]). The following hold for a subset A of a space X: A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 187 a) δ− β -Cl(A) = A ⋃ Int(Cl(δ− Int(A))); b) δ− β -Int(A) = A ⋂ Cl(Int(δ− β − Cl(A))); c) A is δ− β -closed if and only if A = δ− β -Cl(A); d) δ− β -Cl(A) is δ− β -closed; e) X\ δ− β -Cl(A) =δ− β -Int(X\A); f) x ∈ δ− β − Cl(A) if A ⋂ U 6= φ for every δ− β -open set U containing x. Remark 3. From above definitions we have the following diagram in which the converses of implications need not be true, see the examples in [8, 9, 28]. Figure 1: The relationships among some well-known generalized open sets in topological spaces 3. Strong Forms of δ− β-Open Sets In this section we introduce two new strong forms of δ−β -open sets, called δ−β -regular sets and δ − βθ -open sets. By using these sets we introduce a several characterizations of δ− β -open sets and their properties. Definition 1. A subset A of a topological space X is δ− β -regular if it is δ− β -open and δ− β - closed. The family of all δ− β -regular subsets of X containing a point x ∈ X is denoted by δ− βR(X, x), the family of all δ− β -regular sets in X denoted by δ− βR(X, T). Definition 2. A point x of X is called a δ − βθ -cluster point of A if δ − β − Cl(U) ⋂ A 6= φ for every U ∈ δ − βΣ(X , x). The set of all δ − βθ -cluster points of A is called δ − βθ -closure of A and is denoted by δ− β -Clθ (A). A subset A is said to be δ− βθ -closed if A = δ− β -Clθ (A). The complement of a δ− βθ -closed set is said to be δ− βθ -open. Remark 4. The union of two δ−βθ -closed sets is not necessarily δ−βθ -closed as Shown by the following example: Example 1. Let X = {1,2,3}, Define a topology T= {φ, X , {1}, {2}, {1,2}} on X. The subsets {1} and {2} are δ− βθ -closed in (X, T) but {1,2} is not δ− βθ -closed. A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 188 Remark 5. It can be easily shown that δ − β -regular ⇒ δ − βθ -open ⇒ δ − β -open. But the converses are not necessarily true as shown by the following examples: Example 2. Let X = {1,2,3}, Define a topology T= {φ, X , {1}, {2}, {1,2}} on X. The subsets {1,2} is δ− βθ -open in X but not δ− β -regular. Example 3. Let X = {1,2,3,4,5}, Define a topology T= {φ, X , {1}, {3}, {1,3}, {3,4 {1,3,4}} on X.then The subsets {1} is δ− β -open in X but not δ− βθ -open. The following interesting results will play an important role in the sequel. Theorem 1. The following properties hold for a subset A of a topological space (X , T ): a) A∈ δ− βΣ(X , T ) if and only if δ− β -Cl(A) ∈ δ− βR(X , T ); b) A∈ δ− βC(X , T ) if and only if δ− β -Int(A) ∈ δ− βR(X , T ). Proof. a). (Necessity) Let A ∈ δ − βΣ(X , T ),Then we have A ⊂ Cl(Int(δ − Cl(A))) and hence δ− β − Cl(A) ⊂ δ− β − Cl[Cl(Int(δ− Cl(A)))] = Cl(Int(δ− Cl(A))). Since A⊂ δ− β − Cl(A), we have δ− β − Cl(A) ⊂ Cl(Int(δ− Cl(δ− β − Cl(A)))). This shows that δ−β -Cl(A) is a δ−β -open set. On the other hand, δ−β -Cl(A) is always an δ−β -closed set. Therefore δ− β − Cl(A) ∈ δ− βR(X , T ). (Sufficiency). Let δ− β − Cl(A) ∈ δ− βR(X , T ). Then we have A⊂ δ− β − Cl(A) ⊂ Cl(Int(δ− Cl(δ− β − Cl(A)))) ⊂ Cl(Int(δ− Cl(δ− Cl(A)))) =Cl(Int(δ− Cl(A))). Hence we have A⊂ Cl(Int(δ− Cl(A))), There for A∈ δ− βΣ(X , T ). b). This Proof is follows from a) and Lemma (1). Theorem 2. For a subset A of a topological space X; the following are equivalent: a) A∈ δ− βR(X , T ); b) A = δ− β − Cl(δ− β − Int(A)); c) A= δ− β − Int(δ− β − Cl(A)). Proof. The proofs of the implications (a)⇒ (b) and (a)⇒ (c) are obvious thus omitted. (b)⇒ (a) since δ− β -Cl(A) is δ− β -closed, then by Theorem (1) we have δ− β -Int(δ− β − Cl(A)) ∈ δ− βR(X , T ) and A∈ δ− βR(X , T ). (c)⇒ (a) Since δ− β -Int(A) is δ− β -open, then by Theorem (1) we have δ− β − Cl(δ− β − Int(A)) ∈ δ− βR(X , T ) and A∈ δ− βR(X , T ). Theorem 3. For each subset A of a topological space (X, T), we have: δ− β − Clθ (A) = ⋂ {V : A⊂ V and V is δ− βθ − closed} = ⋂ {V : A⊂ V and V ∈ δ− βR(X , T )}. A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 189 Proof. We prove only the first equality since the other is similarly proved. First, suppose that x /∈ δ− β − Clθ (A). Then there exists V ∈ δ− βΣ(X , x) such that δ− β − Cl(V ) ⋂ A= φ. By Theorem 1, X \δ− β -Cl(V) is δ− β -regular and hence X \δ−β–Cl(V) is an δ−βθ -closed set containing A and x /∈ X \δ−β −Cl(V ). Therefore, we have x /∈ ⋂ {V : A⊂ V and V is δ− βθ − closed}. Conversely, suppose that x /∈ ⋂ {V : A⊂ V and V is δ− βθ − closed}. There exists an δ− βθ - closed set V such that A⊂ V and x /∈ V . There exists U ∈ δ− βΣ(X , T ) such that x ∈ U ⊂ δ− β − Cl(U) ⊂ X \ V . Therefore, we have δ− β − Cl(U) ⋂ A⊂ δ− β − Cl(U) ⋂ V = φ. This shows that x /∈ δ− β − Clθ (A). Theorem 4. Let A and B be any two subsets of a topological space (X , T ). Then the following properties hold: a) x ∈ δ− β − Clθ (A) if and only if U ⋂ A 6= φ ; for each U ∈ δ− βR(X , x), b) If A⊂ B; then δ− β − Clθ (A) ⊂ δ− β − Clθ (B), c) δ− β − Clθ (δ− β − Clθ (A)) = δ− β − Clθ (A), d) If Aλ is δ− βθ -closed in X for each λ ∈∆; then ⋂ λ∈∆ Aλ is δ− βθ -closed in X. Proof. The proofs of properties (a) and (b) are obvious, thus omitted. (c) Generally we have δ− β − Clθ (δ− β − Clθ (A)) ⊃ δ− β − Clθ (A). Suppose that x /∈ δ− β − Clθ (A). There exists U ∈ δ− βR(X , x) such that U ⋂ A= φ. Since U ∈ δ− βR(X , T ); we have δ − β − Clθ (A) ⋂ U = φ. This shows that x /∈ δ − β − Clθ (δ − β − Clθ (A)). Therefore, we obtain δ− β − Clθ (δ− β − Clθ (A)) ⊂ δ− β − Clθ (A). (d) Let Aλ is δ − βθ − closed in X for each λ ∈ ∆. For each λ ∈ ∆. Aλ = δ − β − Clθ (Aλ). Hence δ− β − Clθ ( ⋂ λ∈∆ Aλ) ⊂ ⋂ λ∈∆ δ− β − Clθ (Aλ) = ⋂ λ∈∆ Aλ ⊂ δ− β − Clθ ( ⋂ λ∈∆ Aλ). Therefore, we obtain: δ− β − Clθ ( ⋂ λ∈∆ Aλ) = ⋂ λ∈∆ Aλ. This shows that⋂ λ∈∆ Aλ is δ− βθ -closed in X. Corollary 1. Let A and Aλ (λ ∈∆) be any subsets of topological space (X , T ). Then the following properties hold: a) A is δ−βθ -open in X if and only if for each x ∈ A there exists U ∈ δ−βR(X , x) Such that x ∈ U ⊂ A, b) δ− β − Clθ (A) is δ− βθ -closed and δ− β − Intθ (A) is δ− βθ -open, c) If Aλ is δ− βθ -open in X for each λ ∈∆, then ⋃ λ∈∆ Aλ is δ− βθ -open in X. Theorem 5. For a subset A of a space X, the following properties hold: a) If A∈ δ− βΣ(X , T ), Then δ− βCl(A) = δ− β − Clθ (A), b) A∈ δ− βR(X , T ) if and only if A is δ− βθ -open and δ− βθ -closed. A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 190 Proof. (a) Generally we have δ − β − Cl(B) ⊂ δ − β − Clθ (B) for every subset B of X. Let A ∈ δ − βΣ(X , T ). Suppose that x /∈ δ − β − Cl(A).Then there exists U ∈ δ − βΣ(X , x) such that U ⋂ A = φ. Since A ∈ δ − βΣ(X , T ), we have δ − β − Cl(U) ⋂ A = φ. This shows that x /∈ δ − β − Clθ (A). Therefore, we obtain δ − β − Clθ (A) ⊂ δ − β − Cl(A). Hence δ− β − Cl(A) = δ− β − Clθ (A). (b) LetA∈ δ−βR(X , T ). Then A∈ δ−βΣ(X , T ) and by (a), A= δ−β−Cl(A) = δ−β−Clθ (A). Therefore, A is δ − βθ -closed. Since X \ A ∈ δ − βR(X , T ), by the argument above. We have X \ A is δ− βθ -closed and hence A is δ− βθ -open. The converse is obvious 4. Characterizations of Strongly δ− β-Continuous Functions In this section, we introduce some characterizations and basic properties concerning strongly θ −δ− β -continuous functions. Definition 3. A function f : (X , T )→ (Y, T ∗) is said to be strongly θ −δ−β -continuous (briefly, st. θ − δ − β .c.) if for each x ∈ X and each open set V of Y containing f (x), there exists an δ− β -open set U of X containing x, such that f (δ− β − Cl(U)) ⊂ V . Theorem 6. For a function f : (X , T )→ (Y, T ∗), the following are equivalent: a) f is strongly θ −δ− β -continuous, b) For each x ∈ X and each open set V of Y containing f (x), there exists U ∈ δ − βR(X , x) such that f (U) ⊂ V , c) f −1(V ) is δ− βθ -open in X for each open set V of Y, d) f −1(F) is δ− βθ -closed in X for each closed set F of Y, e) f (δ− β − Clθ (A)) ⊂ Cl( f (A)) for each subset A of X, f) δ− β − Clθ ( f −1(B)) ⊂ f −1(Cl(B)) for each subset B of Y. Proof. (a)⇒ (b). It follows directly from Theorem (1). (b) ⇒ (c). Let V be any open set of Y and x ∈ f −1(V ). There exists U ∈ δ − βR(X , x) such that f (U) ⊂ V . Therefore, we have x ∈ U ⊂ f −1(V ). Hence by Corollary (1)(a), f −1(V ) is δ− βθ -open in X. (c)⇒ (d). This is obvious thus omitted. (d)⇒ (e). Let A be any open set of X. since Cl( f (A)) is closed in Y, by (d) f −1(Cl( f (A))) is δ− βθ -closed and we have, δ− β − Clθ (A)) ⊂ δ− β − Clθ ( f −1( f (A))) ⊂ δ− β − Clθ ( f −1(Cl( f (A)))) = f −1(Cl( f (A))). There for, we obtain f (δ− β − Clθ (A)) ⊂ Cl( f (A)). (e)⇒ ( f ). Let B be any subset of Y. By (e), we obtain f (δ−β−Clθ ( f −1(B))) ⊂ Cl( f ( f −1(B))) ⊂ Cl(B) and hence δ−β−Clθ ( f −1(B)) ⊂ f −1(Cl(B)). ( f )⇒ (a). Let x ∈ X and V be any open set of Y containing f (x). Since Y \V is closed in Y, we A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 191 have δ−β−Clθ ( f −1(Y \V )) ⊂ f −1(Cl(Y \V )) = f −1(Y \V ). Therefore, f −1(Y \V ) is δ−βθ - closed in X and f −1(V ) is an δ−βθ -open set containing x. There exists U ∈ δ−βΣ(X , x) such that δ − β − Cl(U) ⊂ f −1(V ) and hence f (δ − β − Cl(U)) ⊂ V . This shows that f is strongly θ −δ− β -continuous. Definition 4 ([9, 27]). A function f : (X , T )→ (Y, T ∗) is said to be δ− β -continuous if f −1(A) is δ− β -open in X for every A∈ T ∗. Theorem 7. Let Y be a regular space. Then f : (X , T )→ (Y, T ∗) is strongly θ −δ−β -continuous if and only if f is δ− β -continuous. Proof. Let x ∈ X and V an open set of Y containing f (x). Since Y is regular, there exists an open set H such that f (x) ∈ H ⊂ Cl(H) ⊂ V . If f is δ− β -continuous, there exists U ∈ δ − βΣ(X , x) such that f (U) ⊂ H. We shall show that f (δ − β − Cl(U)) ⊂ Cl(H). Suppose that y /∈ Cl(H). There exists an open set W containing y such that W ⋂ H = φ. Since f is δ− β -continuous, Then f −1(W ) ∈ δ− βΣ(X , T ) and f −1(W ) ⋂ U = φ. and hence f −1(W ) ⋂ δ− β − Cl(U) = φ. Therefore, we obtain W ⋂ f (δ− β − Cl(U)) = φ and y /∈ f (δ− β − Cl(U)). Consequently, we have f (δ− β − Cl(U)) ⊂ Cl(H) ⊂ V . The converse is obvious. Definition 5. A space X is said to be δ − β -regular if for each closed set F ⊂ X and each point x ∈ X \ F, there exist disjoint δ− β -open sets U and V such that x ∈ U and F ⊂ V . Lemma 2. For a space X the following are equivalent: a) X is δ− β -regular, b) For each point x ∈ X and for each open set U of X containing x, there Exists V ∈ δ−βΣ(X , T ) such that x ∈ V ⊂ δ− β − Cl(V ) ⊂ U, c) For each subset A of X and each closed set F such that A ⋂ F = φ. There exist disjoint U , V ∈ δ− βΣ(X , T ) such that A ⋂ U 6= φ. and F ⊂ V , d) For each closed set F of X, F = ⋂ {δ− β − Cl(V ) : F ⊂ V, V ∈ δ− βΣ(X , T )}. Theorem 8. A continuous function f : (X , T )→ (Y, T ∗) is strongly θ −δ− β -continuous if and only if X is δ− β -regular. Proof. (Necessity). Let f : X → X be the identity function. Then f is continuous and strongly θ −δ−β -continuous by our hypothesis. For any open set U of X and any point x ∈ U , we have f (x) = x ∈ U and there exists V ∈ δ − βΣ(X , x) such that f (δ − β − Cl(V )) ⊂ U . Therefore, we have x ∈ V ⊂ δ−β−Cl(V ) ⊂ U . It follows from Lemma 2 that X is δ−β -regular. (Sufficiency). Suppose that f : X → Y is continuous and X is δ−β -regular. For any x ∈ X and open set V containing f (x), f −1(V ) is an open set containing x. Since X is δ−β -regular, then there exists U ∈ δ−βΣ(X , T ) such that x ∈ U ⊂ δ−β −Cl(U) ⊂ f −1(V ). Therefore, we have f (δ− β − Cl(U)) ⊂ V . This shows that f is strongly θ −δ− β -continuous. A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 192 Theorem 9. Let f : X → Y be a function and g : X → X × Y be the graph function of f. Then, the following properties are hold: a) If g is strongly θ − δ − β -continuous, then f is strongly θ − δ − β -continuous. And X is δ− β -regular. b) If f is strongly θ − δ− β -continuous, and X is δ− β -regular, then g is strongly θ − δ− β - continuous. Proof. (a). Suppose that g is strongly θ − δ − β -continuous. First, we show that f is strongly θ − δ − β -continuous. Let x ∈ X and V an open set of Y containing f (x). Then X × V is an open set of X × Y containing g(x). Since g is strongly θ −δ−β -continuous, then there exists U ∈ δ − βΣ(X , x) such that g(δ − β − Cl(U)) ⊂ X × V . Therefore, we obtain, f (δ− β − Cl(U)) ⊂ V . Next, we show that X is δ− β -regular. Let U be any open set of X and x ∈ U . Since g(x) ∈ U ×Y and U ×Y is an open in X ×Y , Then there exists W ∈ δ−βΣ(X , x) such that g(δ − β − Cl(W )) ⊂ U × Y . Therefore we obtain x ∈W ⊂ δ − β − Cl(W ) ⊂ U and hence X is δ− β -regular. (b). Let x ∈ X and H an open set of X × Y containing g(x). There exists open sets U1 ⊂ X and V ⊂ Y such that g(x) = (x , f (x)) ∈ U1×V ⊂ H. since f is strongly θ −δ−β -continuous, There exists U2 ∈ δ−βΣ(X , x) such that f (δ−β − Cl(U2)) ⊂ V . since X is δ−β -regular and U1 ⋂ U2 ∈ δ− βΣ(X , x), so there exists U ∈ δ− βΣ(X , x) such that x ∈ U ⊂ δ− β − Cl(U) ⊂ U1 ⋂ U2. Therefore we obtain g(δ − β − Cl(U)) ⊂ U1 × f (δ − β − Cl(U2)) ⊂ U1 × V ⊂ H. This show that g is strongly θ −δ− β -continuous. 5. Comparisons and Examples In this section, we investigate the relationships between strongly θ − δ − β -continuous function and other well-known types of strong continuity. Definition 6. A function f : (X , T )→ (Y, T ∗) is said to be: a) Strongly θ -continuous [37] if for each x ∈ X and each open set V of Y Containing f (x), there exists an open set U of X containing x such that f (Cl(U)) ⊂ V; b) Strongly θ -semicontinuous [29] if for each x ∈ X and each open set V of Y Containing f (x), there exists a semi-open set U of X containing x such that f (sC l(U)) ⊂ V; c) Strongly θ -precontinuous [38] if for each x ∈ X and each open set V of Y Containing f (x), there exists a preopen set U of X containing x such that f (pCl(U)) ⊂ V; d) Strongly θ−β -continuous [39] if for each x ∈ X and each open set V of Y Containing f (x), there exists a semi-preopen set U of X containing x such that, f (βCl(U)) ⊂ V; e) Strongly θ -b-continuous [42] if for each x ∈ X and each open set V of Y Containing f (x), there exists a b-open set U of X containing x such that f (bCl(U)) ⊂ V; A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 193 f) Strongly b-continuous [10] if for each x ∈ X and each open set V of Y Containing f (x), there exists U ∈ BO(X , x) such that f (U) ⊂ V; g) Strongly θ -e-continuous [41] if for each x ∈ X and each open set V of Y Containing f (x), there exists an e-open set U of X containing x such that f (e− Cl(U)) ⊂ V . Remark 6. From Definitions 3 and 6 we have the following diagram. However the converses are not true in general by Examples (4.4,4.5,4.6,4.7,4.8) of [42] and (4.2,4.3,4.4,4.5) [41] and the following examples. Figure 2: The relationships between strongly ÎÿâĹŠÎťâŊΚ-continuous functions and other known types of strong continuity Example 4. Let X = {1,2,3,4,5}, Define a topology T = {φ, X , {1}, {3}, {1,3}, {3,4}, {1,3,4}} on X and a topology T ∗ = {φ, X , {4}} on Y. Then the identity function f : X → Y is strongly θ −δ− β -continuous but not strongly θ -b-continuous. Example 5. Let X = {1,2,3,4,5}, Define a topology T = {φ, X , {1}, {3}, {1,3}, {3,4}, {1,3,4}} on X and a topology T ∗ = {φ, X , {2,3,4}} on Y.Then the identity function f : X → Y is strongly θ − b-continuous but not strongly θ -e-continuous. Example 6. Let T be the usual topology for R and T ∗ = {[0,1] ⋃ (1,2) ⋂ Q} where Q denotes the set of rational numbers. Then the identity function f : (R, T )→ (R, T ∗) is strongly θ −δ−β - continuous but neither strongly θ -precontinuous nor strongly θ -semicontinuous. Example 7. Let X = {1,2,3,4}, Define T = {φ, X , {1}, {3}, {1,2}, {1,3}, {1,2,3}, {1,3,4}} on X and a topology T ∗ = {φ, X , {2,4}} on Y.Then the identity function f : (X , T ) → (X , T ∗) is strongly θ −δ−β -continuous but neither strongly θ -e-continuous nor strongly θ −β -continuous. Recall that a space X is said to be submaximal [45] if each dense subset of X is open in X. It is shown in [45] that a space X is submaximal if and only if every preopen set of X is open. A space X is said to be extremally disconnected [3] if the closure of each open set of X is open. Note that an extremally disconnected space is exactly the space where every semiopen set is α-open Theorem 10. Let X be a submaximal extremally disconnected space. Then the following proper- ties are equivalent for a function f : X → Y . A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 194 a) f is strongly θ -continuous; b) f is strongly θ -semicontinuous; c) f is strongly θ -precontinuous; d) f is strongly θ -b-continuous; e) f is strongly θ -e-continuous; f) f is strongly θ − β -continuous; g) f is strongly θ −δ− β -continuous. Proof. it follows from the fact that if X is submaximal extremally disconnected, then open set, preopen set, semiopen set, b-open set, e-open set, semipreopen set and θ −β -open set are equivalent. Theorem 11. Let f : (X , T )→ (Y, T ∗) and g : (Y, T ∗)→ (Z , T ∗∗) be a functions. If f is strongly θ−δ−β -continuous. And g is continuous, then the composition function go f : (X , T )→ (Z , T ∗∗) is strongly θ −δ− β -continuous. Proof. This proof follows directly from Theorem 6. 6. Strongly θ −δ− β-continuous Functions and Separation Axioms Definition 7 ([7, 28]). A space X is said to be δ−β − T2 if for each pair of distinct points x and y in X, there exist U ∈ δ− βΣ(X , x) and V ∈ δ− βΣ(X , y) such that U ⋂ V = φ. Lemma 3. A space X is δ−β − T2 if and only if for each pair of distinct points x and y in X, there exist U ∈ δ− βΣ(X , x) and V ∈ δ− βΣ(X , y) such that δ− β − Cl(U) ⋂ δ− β − Cl(V ) = φ. Theorem 12. If a function f : (X , T )→ (Y, T ∗) is (st. θ − δ− β .c.) injection and Y is T0, then X is δ− β − T2. Proof. For any distinct points x and y of X, by hypothesis f (x) 6= f (y) and there exists either an open set V containing f (x) not containing f (y) or an open set H containing f (y) not containing f (x). If the first case holds, then there exists U ∈ δ − βΣ(X , x) such that f (δ− β − Cl(U)) ⊂ V . Thus, we obtain f (y) /∈ f (δ− β − Cl(U)) and hence X \ δ − β − Cl(U) ∈ δ − βΣ(X , y). If the second case holds, then we obtain a similar result. Thus, X is δ− β − T2. Theorem 13. If f : (X , T ) → (Y, T ∗) (st. θ − δ − β .c.)function and Y is Hausdorff, then the subset A= {(x , y) : f (x) = f (y)} is δ− βθ -closed in X × X . A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 195 Proof. It is clear that f (x) 6= f (y) for each (x , y) /∈ A. Since Y is Hausdorff, there exist open sets V and H of Y containing f (x) and f (y), respectively, such that V ⋂ H = φ. Since f is (st. θ−δ−β .c). There exist U ∈ δ−βΣ(X , x) and W ∈ δ−βΣ(X , y) such that f (δ−β−Cl(U)) ⊂ V and f (δ − β − Cl(W )) ⊂ H. Set D = f (δ − β − Cl(U))× f (δ − β − Cl(W )). It follows that (x , y) ∈ D ∈ δ − βR(X × X ) and D ⋂ A = φ. This means δ − β − Clθ (A) ⊂ A and thus, A is δ− βθ -closed in X × X . Recall that for a function f : X → Y , the subset {(x , f (x)) : x ∈ X } of X × Y is called the graph of f and is denoted by G( f ). Definition 8. The graph G( f ) of a function f : X → Y is said to be strongly δ − β -closed if for each (x , y) ∈ (X × Y ) \ G( f ); there exist U ∈ δ− βΣ(X , x) and an open set V in Y containing y such that (δ− β − Cl(U)× V ) ⋂ G( f ) = φ. Lemma 4. The graph G( f ) of a function f : X → Y is strongly δ − β -closed if and only if for each (x , y) ∈ (X × Y ) \ G( f ), there exist U ∈ δ− βΣ(X , x) and an open set V in Y containing y such that f (δ− β − Cl(U)) ⋂ V = φ. Theorem 14. If f : X → Y is (st. θ − δ − β .c.) and Y is Hausdorff, then G(f) is strongly δ− β -closed in X × Y . Proof. It is clear that f (x) 6= y for each (x , y) ∈ (X ×Y )\G( f ). Since Y is Hausdorff, there exist open sets V and H in Y containing f (x) and y, respectively, such that V ⋂ H = φ. Since f is (st. θ − δ − β .c.), There exist U ∈ δ − βΣ(X , x) such that f (δ − β − Cl(U)) ⊂ V . Thus, f (δ−β −Cl(U)) ⋂ H = φ and then by Lemma (4),G( f ) is strongly δ−β -closed in X ×Y . 7. Covering Properties Definition 9. A space X is said to be: a) δ − β -closed if every cover of X by δ − β -open sets has a finite subcover whose preclosures cover X, b) Countably δ−β -closed if every countable cover of X by δ−β -open sets has a Finite subcover whose preclosures cover X. A subset K of a space X is said to be δ− β -closed relative to X if for every cover {Vλ : λ ∈∆} of K by δ− β -open sets of X, there exists a finite subset ∆0 of ∆ such that K ⊂ ⋃ {δ− β − Cl(Vλ) : λ ∈∆0}. Theorem 15. If f : X → Y is (st. θ −δ−β .c.) and K is δ−β -closed relative to X, then, f(K) is a compact set of Y. Proof. Suppose that f : X → Y is (st. θ −δ− β .c.) and K is δ− β -closed relative to X, Let {Vλ : λ ∈∆} be a cover of f (K) by open sets of Y. For each point x ∈ K , there exists λ(x) ∈∆ A. M. Farhan and X. Yang / Eur. J. Pure Appl. Math, 8 (2015), 185-200 196 such that f (x) ∈ Vλ(x). Since f is (st. θ − δ− β .c.), there exists Ux ∈ δ− βΣ(X , x) such that f (δ−β −Cl(Ux)) ⊂ Vλ(x). The family {Ux : x ∈ K} is a cover of K by δ−β -open sets of X and hence there exists a finite subset K0 of K such that K ⊂ ⋃ x∈ko δ− β − Cl(Ux). Therefore, we obtain f (K) ⊂ ⋃ x∈ko Vα(x). This shows that f (K) is compact. Corollary 2. Let f : X → Y is (st. θ −δ− β .c.) surjection. Then the following Properties hold: a) If X is δ− β -closed, then Y is compact, b) If X is countably δ− β -closed, then Y is countably compact. Theorem 16. If a function f : X → Y has a strongly δ−β -closed graph, then f(K) is closed in Y for each subset K which is δ− β -closed relative to X. Proof. Let K be δ − β -closed relative to X and y ∈ Y \ f (K). Then for each x ∈ K we have (x , y) /∈ G( f ) and by Lemma 4 there exist Ux ∈ δ − βΣ(X , x) and an open set Vx of Y containing y such that f (δ − β − Cl(Ux)) ⋂ Vx = φ. The family {Ux : x ∈ K} is a cover of K by δ− β -open sets of X. Since K is δ− β -closed relative to X, there exists a finite subset K0 of K such that K ⊂ ⋃ {δ− β − Cl(Ux) : x ∈ K0}. Put V = ⋂ {Vx : x ∈ K0}. Then V is an open set containing y and f (K) ⋂ V ⊂ [ ⋃ x∈ko f (δ− β − Cl(Ux))] ⋂ V ⊂ ⋃ x∈ko [ f (δ− β − Cl(Ux)) ⋂ Vx] = φ. Therefore, we have y /∈ Cl( f (K)) and hence f (K) is closed in Y. Theorem 17. Let X be a submaximal extremally disconnected space. If a function f : X → Y has a strongly δ− β -closed graph, then f −1(K) is θ -closed in X for each compact set K of Y. Proof. Let K be a compact set of Y and x /∈ f −1(K). Then for each y ∈ K we have (x , y) /∈ G( f ) and by Lemma 4 there exist Uy ∈ δ−βΣ(X , x) and an open set Vy of Y containing y such that f (δ − β − Cl(Uy)) ⋂ Vy = φ. The family {Vy : y ∈ K} is an open cover of K and there exists a finite subset K0 of K such that K ⊂ ⋃ y∈ko Vy . Since X is submaximal extremally disconnected, therefor each Uy is open in X and δ−β −Cl(Uy) = Cl(Uy). Set U = ⋂ y∈ko Uy , then U is an open set containing x and f (Cl(U)) ⋂ K ⊂ ⋃ x∈ko [ f (Cl(U)) ⋂ Vy] ⊂ ⋃ x∈ko [ f (δ− β − Cl(Uy)) ⋂ Vy] = φ. so, we have Cl(U) ⋂ f −1(K) = φ and hence x /∈ Clθ ( f −1(K)). This shows that f −1(K) is θ -closed in X. Corollary 3. Let X be a submaximal extremally disconnected space and Y be a compact Hausdorff space. For a function f : X → Y , the following properties are equivalent: a) f is (st. θ −δ− β .c.); b) G( f ) is strongly δ− β -closed in X × Y ; c) f is strongly θ -continuous; d) f is continuous; REFERENCES 197 e) f is δ− β -continuous. Proof. (a)⇒ (b). It follows directly from Theorem 14. (b)⇒ (c). It follows directly from Theorem 17. (c)⇒ (d)⇒ (e). These are clear. (e)⇒ (a). Since Y is regular, it follows from Theorem 7. 8. Conclusion Topology as a field of mathematics is concerned with all questions directly or indirectly related to continuity. Therefore, generalization of continuity is one of the most important sub- jects in topology. One of the most important subjects in studying topology and physics is conti- nuity, has been researched and investigated by many mathematicians and quantum physicists. [4–6, 17, 22, 40, 43, 44] from the different points of views. Relation of topology and physics have been appeared in [17, 18, 26, 47], El-Naschie in [13, 17, 26] have indicate that topology plays a significant role in quantum physics, high energy physics and superstring theory. One can observe the influence made in the realms of applied research by general topological spaces, properties and structures. In digital topology, information systems, particle physic [30], com- putational topology for geometric design and molecular design [35], Furthermore, Rosen and Peters [46] have used topology in computer-aided geometric design and engineering design. 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