@ Appl. Gen. Topol. 15, no. 1 (2014), 93-109doi:10.4995/agt.2014.2268 c© AGT, UPV, 2014 Soft set theory and topology D. N. Georgiou a and A. C. Megaritis b a Department of Mathematics, University of Patras, 265 04 Patras, Greece. (georgiou@math.upatras.gr) b Technological Educational Institute of Western Greece, Greece. (thanasismeg13@gmail.com) Abstract In this paper we study and discuss the soft set theory giving new defi- nitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets. Furthermore, we investigate the theory of soft topological spaces and we present new definitions, charac- terizations, and properties concerning the soft closure, the soft interior, the soft boundary, the soft continuity, the soft open and closed maps, and the soft homeomorphism. 2010 MSC: 54A05; 06D72. Keywords: Soft set theory; Soft topology. 1. Preliminaries For every set X we denote by P(X) the power set of X, that is the set of all subsets of X and by |X| the cardinality of X. Also, we denote by ω the first infinite cardinal and by R the set of real numbers. In 1999 D. Molodtsov (see [17]) introduced the notion of soft set. Later, he applied this theory to several directions (see [18], [19], and [20]). The soft set theory has been applied to many different fields (see, for exam- ple, [1], [2], [4], [5], [7], [8], [10], [12], [13], [14], [15], [21], [23], [25]). In 2011 and 2012 few researches (see, for example, [3], [6], [9], [16], [22], [24]) introduced and studied the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. The paper is organized as follows. In section 2 we study and discuss the soft set theory giving new definitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets. In section 3 we Received July 2012 – Accepted October 2012 http://dx.doi.org/10.4995/agt.2014.2268 D. N. Georgiou and A. C. Megaritis investigate the theory of soft topological spaces and we present new definitions, characterizations, and many properties concerning the soft closure, the soft interior, the soft boundary, the soft continuity, the soft open and closed maps, and the soft homeomorphism. 2. Soft Set Theory Definition 2.1 (see [17]). Let X be an initial universe set and A a set of parameters. A pair (F, A), where F is a map from A to P(X), is called a soft set over X. In what follows by SS(X, A) we denote the family of all soft sets (F, A) over X. Definition 2.2 (see [17]). Let (F, A), (G, A) ∈ SS(X, A). We say that the pair (F, A) is a soft subset of (G, A) if F(p) ⊆ G(p), for every p ∈ A. Symbolically, we write (F, A) ⊑ (G, A). Also, we say that the pairs (F, A) and (G, A) are soft equal if (F, A) ⊑ (G, A) and (G, A) ⊑ (F, A). Symbolically, we write (F, A) = (G, A). Definition 2.3 (see, for example, [17] and [24]). Let I be an arbitrary index set and {(Fi, A) : i ∈ I} ⊆ SS(X, A). The soft union of these soft sets is the soft set (F, A) ∈ SS(X, A), where the map F : A → P(X) defined as follows: F(p) = ∪{Fi(p) : i ∈ I}, for every p ∈ A. Symbolically, we write (F, A) = ⊔{(Fi, A) : i ∈ I}. Example 2.4. Let X = R, A = {0, 1}, and I = {1, 2, . . .}. For every i ∈ I we consider the soft set (Fi, A), where the map Fi : A → P(X) defined as follows: Fi(p) = { (0, i), if p = 0, (−i, 0), if p = 1. Then, ⊔{(Fi, A) : i ∈ I} = (F, A), where the map F : A → P(X) defined as follows: F(p) = { (0, +∞), if p = 0, (−∞, 0), if p = 1. Definition 2.5 (see, for example, [17] and [24]). Let I be an arbitrary index set and {(Fi, A) : i ∈ I} ⊆ SS(X, A). The soft intersection of these soft sets is the soft set (F, A) ∈ SS(X, A), where the map F : A → P(X) defined as follows: F(p) = ∩{Fi(p) : i ∈ I}, for every p ∈ A. Symbolically, we write (F, A) = ⊓{(Fi, A) : i ∈ I}. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 94 Soft set theory and topology Example 2.6. Let X = R, A = {0, 1, 2}, and I = {1, 2, . . .}. For every i ∈ I we consider the soft set (Fi, A), where the map Fi : A → P(X) defined as follows: Fi(p) =      (−1 i , 1 i ), if p = 0, (1 − 1 i , 1 + 1 i ), if p = 1, (2 − 1 i , 2 + 1 i ), if p = 2. Then, ⊓{(Fi, A) : i ∈ I} = (F, A), where the map F : A → P(X) defined as follows: F(p) =      {0}, if p = 0, {1}, if p = 1, {2}, if p = 2. Definition 2.7 (see, for example, [24]). Let (F, A) ∈ SS(X, A). The soft com- plement of (F, A) is the soft set (H, A) ∈ SS(X, A), where the map H : A → P(X) defined as follows: H(p) = X \ F(p), for every p ∈ A. Symbolically, we write (H, A) = (F, A)c. Example 2.8. Let X = R and A = {1, 2, . . .}. We consider the soft set (F, A), where the map F : A → P(X) defined as follows: F(p) = [p, +∞), for every p ∈ A. Then, (F, A)c = (H, A), where the map H : A → P(X) defined as follows: H(p) = (−∞, p), for every p ∈ A. Definition 2.9 (see [17]). The soft set (F, A) ∈ SS(X, A), where F(p) = ∅, for every p ∈ A is called the A-null soft set of SS(X, A) and denoted by 0A. The soft set (F, A) ∈ SS(X, A), where F(p) = X, for every p ∈ A is called the A-absolute soft set of SS(X, A) and denoted by 1A. The proofs of the following propositions are straightforward verifications of the above definitions. Proposition 2.10. Let (F, A) ∈ SS(X, A). The following statements are true: (1) (F, A) ⊓ (F, A) = (F, A). (2) (F, A) ⊔ (F, A) = (F, A). (3) (F, A) ⊓ 0A = 0A. (4) (F, A) ⊔ 0A = (F, A). (5) (F, A) ⊓ 1A = (F, A). (6) (F, A) ⊔ 1A = 1A. (7) (F, A) ⊓ (F, A)c = 0A. (8) (F, A) ⊔ (F, A)c = 1A. (9) (0A) c = 1A. (10) (1A) c = 0A. (11) ((F, A)c)c = (F, A). (12) 0A ⊑ (F, A) ⊑ 1A. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 95 D. N. Georgiou and A. C. Megaritis Proposition 2.11. Let (F, A), (G, A), (H, A) ∈ SS(X, A). The following state- ments are true: (1) (F, A) ⊓ ((G, A) ⊓ (H, A)) = ((F, A) ⊓ (G, A)) ⊓ (H, A). (2) (F, A) ⊔ ((G, A) ⊔ (H, A)) = ((F, A) ⊔ (G, A)) ⊔ (H, A). (3) (F, A) ⊓ ((G, A) ⊔ (H, A)) = ((F, A) ⊓ (G, A)) ⊔ ((F, A) ⊓ (H, A)). (4) (F, A) ⊔ ((G, A) ⊓ (H, A)) = ((F, A) ⊔ (G, A)) ⊓ ((F, A) ⊔ (H, A)). Proposition 2.12. Let I be an arbitrary set and {(Fi, A) : i ∈ I} ⊆ SS(X, A). The following statements are true: (1) (Fi, A) ⊑ ⊔{(Fi, A) : i ∈ I}, for every i ∈ I. (2) ⊓{(Fi, A) : i ∈ I} ⊑ (Fi, A), for every i ∈ I. (3) (⊔{(Fi, A) : i ∈ I}) c = ⊓{(Fi, A) c : i ∈ I}. (4) (⊓{(Fi, A) : i ∈ I}) c = ⊔{(Fi, A) c : i ∈ I}. Definition 2.13. Let (F, A), (G, A) ∈ SS(X, A). The soft symmetric difference of these soft sets is the soft set (H, A) ∈ SS(X, A), where the map H : A → P(X) defined as follows: H(p) = (F(p)\G(p))∪(G(p)\F(p)), for every p ∈ A. Symbolically, we write (H, A) = (F, A) △ (G, A). Example 2.14. Let X = {1, 2, 3, 4, 5} and A = {0, 1, 2, . . .}. We consider the soft sets (F, A) and (G, A), where the maps F : A → P(X) and G : A → P(X) defined as follows: F(p) = { {1, 2, 3, 4}, if p = 0, ∅, otherwise, G(p) = { {1, 4, 5}, if p = 0, ∅, otherwise. Then, (F, A) △ (G, A) = (H, A), where the map H : A → P(X) defined as follows: H(p) = { {2, 3, 5}, if p = 0, ∅, otherwise. The proof of the following proposition is straightforward verification of the Definition 2.13. Proposition 2.15. Let (F, A), (G, A), (H, A) ∈ SS(X, A). The following state- ments are true: (1) (F, A) △ ((G, A) △ (H, A)) = ((F, A) △ (G, A)) △ (H, A). (2) (F, A) △ (G, A) = (G, A) △ (F, A). (3) (F, A) △ 0A = (F, A). (4) (F, A) △ (F, A) = 0A. (5) (F, A) ⊓ ((G, A) △ (H, A)) = ((F, A) ⊓ (G, A)) △ ((F, A) ⊓ (H, A)). Remark 2.16. By Proposition 2.15 follows that the pair (SS(X, A), △) is a group of soft sets. The identity element is the soft set 0A and the inverse of the element (F, A) is the soft set (F, A). Also, the triad (SS(X, A), △, ⊓) is a ring of soft sets. Let X and Y be two initial universe sets, PX and PY two sets of parameters, f : X → Y , and e : PX → PY . In [11] the authors, using f and e, define the c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 96 Soft set theory and topology notion of a mapping from the family of all soft sets (F, A) over X, where A ⊆ PX, to the family of all soft sets (G, B) over Y , where B ⊆ PY . In [24] the authors gave a mapping from SS(X, A) to SS(Y, B) and studied properties of images and inverse images of soft sets. The given below definition is actually the definition of this mapping. Definition 2.17. Let X and Y be two initial universe sets, A and B two sets of parameters, f : X → Y , and e : A → B. Then, by Φfe we denote the map from SS(X, A) to SS(Y, B) for which: (1) If (F, A) ∈ SS(X, A), then the image of (F, A) under Φfe, denoted by Φfe(F, A), is the soft set (G, B) ∈ SS(Y, B) such that G(pY ) = { ⋃ {f(F(p)) : p ∈ e−1({pY })}, if e −1({pY }) 6= ∅, ∅, if e−1({pY }) = ∅, for every pY ∈ B. (2) If (G, B) ∈ SS(Y, B), then the inverse image of (G, B) under Φfe, denoted by Φ−1 fe (G, B), is the soft set (F, A) ∈ SS(X, A) such that F(pX) = f −1(G(e(pX))), for every pX ∈ A. The following propositions are easily proved. Proposition 2.18. Let (F, A), (F1, A) ∈ SS(X, A), (G, B), (G1, B) ∈ SS(Y, B). The following statements are true: (1) If (F, A) ⊑ (F1, A), then Φfe(F, A) ⊑ Φfe(F1, A). (2) If (G, B) ⊑ (G1, B), then Φ −1 fe (G, B) ⊑ Φ−1 fe (G1, B). (3) (F, A) ⊑ Φ−1 fe (Φfe(F, A)). (4) If f is an 1-1 map of X into Y and e is an 1-1 map of A into B, then (F, A) = Φ−1 fe (Φfe(F, A)). (5) Φfe(Φ −1 fe (G, B)) ⊑ (G, B). (6) If f is a map of X onto Y and e is a map of A onto B, then Φfe(Φ −1 fe (G, B)) = (G, B). (7) Φ−1 fe ((G, B)c) = (Φ−1 fe (G, B))c. Proposition 2.19. Let I be an arbitrary set, {(Fi, A) : i ∈ I} ⊆ SS(X, A), and {(Gi, B) : i ∈ I} ⊆ SS(Y, B). The following statements are true: (1) Φfe(⊔{(Fi, A) : i ∈ I}) = ⊔{Φfe(Fi, A) : i ∈ I}. (2) Φfe(⊓{(Fi, A) : i ∈ I}) ⊑ ⊓{Φfe(Fi, A) : i ∈ I}. (3) Φ−1 fe (⊔{(Gi, B) : i ∈ I}) = ⊔{Φ −1 fe (Gi, B) : i ∈ I}. (4) Φ−1 fe (⊓{(Gi, B) : i ∈ I}) = ⊓{Φ −1 fe (Gi, B) : i ∈ I}. Definition 2.20. Define the order of a soft set (F, A) ∈ SS(X, A) as follows: (1) ord(F, A) = n, where n ∈ ω, if and only if the intersection of any n + 2 distinct elements of {F(p) : p ∈ A} is empty and there exist n + 1 distinct elements of {F(p) : p ∈ A}, whose intersection is not empty. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 97 D. N. Georgiou and A. C. Megaritis (2) ord(F, A) = ∞, if and only if for every n ∈ ω there exist n distinct elements of {F(p) : p ∈ A}, whose intersection is not empty. Note. Let X be an initial universe set and A a set of parameters. We consider the following subsets of SS(X, A): (1) C(X, A) = {(F, A) ∈ SS(X, A) : ⋃ {F(p) : p ∈ A} = X}. (2) SS(X, A, ν) = {(F, A) ∈ SS(X, A) : |F(p)| = ν, for every p ∈ A}, where ν is an ordinal such that ν ≤ |X|. (3) F(X, A) = {(F, A) ∈ SS(X, A) : |F(p)| < ω, for every p ∈ A}. (4) O(X, A, n) = {(F, A) ∈ SS(X, A) : ord(F, A) = n}, where n ∈ ω∪{∞}. Example 2.21. (1) Let X be a nonempty set. Every cover {Ui : i ∈ I} of X, that is ∪{Ui : i ∈ I} = X, can be considered as the soft set (F, A) ∈ C(X, A), where A = I and the map F : A → P(X) defined as follows: F(i) = Ui, for every i ∈ A. (2) Let X be a set with |X| = 5. Then, the family of all subsets Y of X with |Y | = 3 can be considered as the element (F, A) of SS(X, A, 3), where A = {1, 2, . . . , 10} and F is an 1-1 map of A to P(X). Proposition 2.22. Let (F, A) ∈ SS(X, A) and (G, B) ∈ SS(Y, B). The follow- ing statements are true: (1) If (F, A) ∈ C(X, A) and the maps f : X → Y and e : A → B are onto, then Φfe(F, A) ∈ C(Y, B). (2) If (F, A) ∈ SS(X, A, ν), the map f : X → Y is 1-1, and the map e : A → B is 1-1 and onto, then Φfe(F, A) ∈ SS(Y, B, ν). (3) If (F, A) ∈ F(X, A), the map f : X → Y is 1-1, and the map e : A → B is 1-1 and onto, then Φfe(F, A) ∈ F(Y, B). (4) If (F, A) ∈ O(X, A, n), the map f : X → Y is 1-1, and the map e : A → B is 1-1 and onto, then Φfe(F, A) ∈ O(Y, B, n). (5) If (G, B) ∈ C(Y, B), then Φ−1 fe (G, B) ∈ C(X, A). (6) If (G, B) ∈ SS(Y, B, ν) and the map f : X → Y is 1-1 and onto, then Φ−1 fe (G, B) ∈ SS(X, A, ν). (7) If (G, B) ∈ F(Y, B) and the map f : X → Y is 1-1, then Φ−1 fe (G, B) ∈ F(X, A). (8) If (G, B) ∈ O(Y, B, n) and the map f : X → Y is onto, then Φ−1 fe (G, B) ∈ O(X, A, n). Proof. Suggestively we prove the statements (1), (2), (7), and (8). (1) Let (F, A) ∈ C(X, A) and Φfe(F, A) = (G, B). Then, ⋃ {F(pX) : pX ∈ A} = X. Since the map f : X → Y is onto, f(X) = Y . Therefore, ⋃ {G(pY ) : pY ∈ B} = ⋃ { ⋃ {f(F(p)) : p ∈ e−1({pY })} : pY ∈ B} = ⋃ {f(F(pX)) : pX ∈ A} = f(∪{F(pX) : pX ∈ A}) = f(X) = Y. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 98 Soft set theory and topology Thus, Φfe(F, A) ∈ C(Y, B). (2) Let (F, A) ∈ SS(X, A, ν) and Φfe(F, A) = (G, B). Then, |F(p)| = ν, for every p ∈ A. Let pY ∈ B. Since the map e : A → B is 1-1, G(pY ) = f(F(p)), where p ∈ e−1({pY }). Also, since the map f : X → Y is 1-1, we have |f(F(p))| = |F(p)|. Therefore, |G(pY )| = |f(F(p))| = |F(p)| = ν. Thus, Φfe(F, A) ∈ SS(Y, B, ν). (7) Let (G, B) ∈ F(Y, B) and Φ−1 fe (G, B) = (F, A). Then, |G(pY )| < ω, for every pY ∈ B. Let pX ∈ A. Then, F(pX) = f −1(G(e(pX))). Since the map f : X → Y is 1-1, we have |F(pX)| = |f −1(G(e(pX)))| < ω. This means that Φ−1 fe (G, B) ∈ F(X, A). (8) Let (G, B) ∈ O(Y, B, n) and Φ−1 fe (G, B) = (F, A). Then, the intersection of any n + 2 distinct elements of {G(pY ) : pY ∈ B} is empty and there exist n + 1 distinct elements of {G(pY ) : pY ∈ B}, whose intersection is not empty. Let p1X, . . . , p n+1 X ∈ A such that G(e(p1X)) ∩ . . . ∩ G(e(p n+1 X )) 6= ∅. Then, F(p1X) ∩ . . . ∩ F(p n+1 X ) = f−1(G(e(p1X))) ∩ . . . ∩ f −1(G(e(pn+1 X ))) = f−1(G(e(p1X)) ∩ . . . ∩ G(e(p n+1 X ))). Since the map f : X → Y is onto, F(p1X) ∩ . . . ∩ F(p n+1 X ) 6= ∅. This means that there exist n+1 distinct elements of {F(pX) : pX ∈ A}, whose intersection is not empty. Now, we prove that the intersection of any n + 2 distinct elements of the set {F(pX) : pX ∈ A} is empty. Let p 1 X, . . . , p n+2 X ∈ A. Then, F(p1X) ∩ . . . ∩ F(p n+2 X ) = f−1(G(e(p1X))) ∩ . . . ∩ f −1(G(e(pn+2 X ))) = f−1(G(e(p1X)) ∩ . . . ∩ G(e(p n+2 X ))) = f−1(∅) = ∅. Thus, Φ−1 fe (G, B) ∈ O(X, A, n). � 3. Soft Topology Definition 3.1 (see, for example, [24]). Let X be an initial universe set, A a set of parameters, and τ ⊆ SS(X, A). We say that the family τ defines a soft topology on X if the following axioms are true: (1) 0A, 1A ∈ τ. (2) If (G, A), (H, A) ∈ τ, then (G, A) ⊓ (H, A) ∈ τ. (3) If (Gi, A) ∈ τ for every i ∈ I, then ⊔{(Gi, A) : i ∈ I} ∈ τ. The triplet (X, τ, A) is called a soft topological space or soft space. The members of τ are called soft open sets in X. Also, a soft set (F, A) is called soft closed if the complement (F, A)c belongs to τ. The family of soft closed sets is denoted by τc. Remark 3.2. Let (X, τ, A) be a soft topological space. Then, by Proposition 2.10 (11) we have (G, A) ∈ τ if and only if (G, A)c ∈ τc. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 99 D. N. Georgiou and A. C. Megaritis The proof of the following proposition is straightforward verification of the Definition 3.1, Proposition 2.10, and Proposition 2.12. Proposition 3.3. Let (X, τ, A) be a soft topological space. The family τc has the following properties: (1) 0A, 1A ∈ τ c. (2) If (Q, A), (R, A) ∈ τc, then (Q, A) ⊔ (R, A) ∈ τc. (3) If (Qi, A) ∈ τ c for every i ∈ I, then ⊓{(Qi, A) : i ∈ I} ∈ τ c. Example 3.4. (1) Let X = {1, 2, . . .}, A = {0, 1}, and τ = {(Gn, A) : n = 1, 2, . . .} ∪ {0A, 1A}, where the map Gn : A → P(X) defined as follows: Gn(p) = { {n, n + 1, . . .}, if p = 0, ∅, if p = 1. The triplet (X, τ, A) is a soft topological space. (2) Let (X, t) be a topological space, A a nonempty set, and τ = {(GU, A) : U ∈ t}, where the map GU : A → P(X) defined as follows: GU (p) = U, for every p ∈ A. The triplet (X, τ, A) is a soft topological space. Definition 3.5. Let (X, τ, A) be a soft topological space, a ∈ A, and x ∈ X. We say that a soft set (F, A) ∈ τ is an a-soft open neighborhood of x in (X, τ, A) if x ∈ F(a). Proposition 3.6. Let (X, τ, A) be a soft topological space. Then, (G, A) ∈ τ if and only if for every a ∈ A and x ∈ G(a) there exists an a-soft open neighborhood (G(a,x), A) of x in (X, τ, A) such that (G(a,x), A) ⊑ (G, A). Proof. If (G, A) ∈ τ, then for every a ∈ A and x ∈ G(a) we consider the soft set (G(a,x), A), where G(a,x) = G. Obviously, (G(a,x), A) is an a-soft open neighborhood of x. Conversely, we suppose that for every a ∈ A and x ∈ G(a) there exists an a-soft open neighborhood (G(a,x), A) of x in (X, τ, A) such that (G(a,x), A) ⊑ (G, A), that is G(a,x)(p) ⊆ G(p), for every p ∈ A. (1) We prove that (G, A) ∈ τ. We set I = {(a, x) : a ∈ A, x ∈ G(a)}. It suffices to prove that (G, A) = ⊔{(G(a,x), A) : (a, x) ∈ I} or equivalently G(p) = ∪{G(a,x)(p) : (a, x) ∈ I}, for every p ∈ A. Let p ∈ A. By relation (1) we have G(a,x)(p) ⊆ G(p), for every (a, x) ∈ I. Therefore, ∪{G(a,x)(p) : (a, x) ∈ I} ⊆ G(p). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 100 Soft set theory and topology We prove that G(p) ⊆ ∪{G(a,x)(p) : (a, x) ∈ I}. Let y ∈ G(p). Then, by assumption there exists a p-soft open neighborhood (G(p,y), A) of y in (X, τ, A) such that (G(p,y), A) ⊑ (G, A). Therefore, y ∈ G(p,y)(p) ⊆ ∪{G(a,x)(p) : (a, x) ∈ I}. Thus, (G, A) = ⊔{(G(a,x), A) : (a, x) ∈ I}. � Definition 3.7 (see [22]). Let (X, τ, A) be a soft topological space. The soft closure Cl(F, A) of (F, A) ∈ SS(X, A) is the soft set ⊓{(Q, A) ∈ τc : (F, A) ⊑ (Q, A)}. Definition 3.8. Let (X, τ, A) be a soft topological space and a ∈ A. A point x ∈ X is said to be an a-cluster point of (F, A) ∈ SS(X, A) if for every a-soft open neighborhood (G, A) of x we have (F, A) ⊓ (G, A) 6= 0A. The set of all a-cluster points of (F, A) is denoted by cl(F, a). Also, the set of all a-cluster points of (F, A)c is denoted by cl((F, a)c). Proposition 3.9. Let (X, τ, A) be a soft space and (F, A) ∈ SS(X, A). Then, Cl(F, A) = (RF,A, A), where the map RF,A : A → P(X) defined as follows: RF,A(p) = F(p) ∪ cl(F, p), for every p ∈ A. Proof. We need to prove that (a) (F, A) ⊑ (RF,A, A), (b) (RF,A, A) ∈ τ c, and (c) (RF,A, A) ⊑ (Q, A), for every (Q, A) ∈ τ c such that (F, A) ⊑ (Q, A). (a) First we observe that F(p) ⊆ F(p) ∪ cl(F, p) = RF,A(p), for every p ∈ A. Thus, (F, A) ⊑ (RF,A, A). (b) We prove that (RF,A, A) ∈ τ c or equivalently (RF,A, A) c ∈ τ. Let a ∈ A and x ∈ X \ RF,A(a) = X \ (F(a) ∪ cl(F, a)). By Proposition 3.6, it suffices to prove that there exists an a-soft open neighborhood (G(a,x), A) of x such that (G(a,x), A) ⊑ (RF,A, A) c or equivalently G(a,x)(p) ⊆ X \ RF,A(p) = X \ (F(p) ∪ cl(F, p)), for every p ∈ A. Since x /∈ cl(F, a), there exists an a-soft open neighborhood (G(a,x), A) of x such that (F, A) ⊓ (G(a,x), A) = 0A. This means that F(p) ∩ G(a,x)(p) = ∅, for every p ∈ A. Therefore, G(a,x)(p) ⊆ X \ F(p), for every p ∈ A. We prove that G(a,x)(p) ⊆ X \ cl(F, p), for every p ∈ A. Indeed, let y ∈ G(a,x)(p), where p ∈ A. Then, the soft set (G(a,x), A) is a p-soft open neigh- borhood of y such that (F, A)⊓(G(a,x), A) = 0A and, therefore, y ∈ X \cl(F, p). Thus, G(a,x)(p) ⊆ (X \ F(p)) ∩ (X \ cl(F, p)) = X \ (F(p) ∪ cl(F, p)), for every p ∈ A. (c) Finally, let (Q, A) ∈ τc such that (F, A) ⊑ (Q, A). (2) We prove that (RF,A, A) ⊑ (Q, A). Since RF,A(p) = F(p)∪cl(F, p) and F(p) ⊆ Q(p), for every p ∈ A, it suffices to prove that cl(F, p) ⊆ Q(p) or X \ Q(p) ⊆ X \ cl(F, p), for every p ∈ A. Indeed, let y ∈ X \ Q(p) and y ∈ cl(F, p), where p ∈ A. We observe that the soft set (Q, A)c is a p-soft open neighborhood c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 101 D. N. Georgiou and A. C. Megaritis of y such that (F, A) ⊓ (Q, A)c 6= 0A which contradicts relation (2). Thus, y ∈ X \ cl(F, p). � Definition 3.10 (see [24]). Let (X, τ, A) be a soft topological space. The soft interior Int(F, A) of (F, A) ∈ SS(X, A) is the soft set ⊔{(G, A) ∈ τ : (G, A) ⊑ (F, A)}. Definition 3.11. Let (X, τ, A) be a soft topological space and a ∈ A. A point x ∈ X is said to be an a-interior point of (F, A) ∈ SS(X, A) if there exists an a-soft open neighborhood (G, A) of x such that (G, A) ⊑ (F, A). The set of all a-interior points of (F, A) is denoted by int(F, a). Proposition 3.12. Let (X, τ, A) be a soft space and (F, A) ∈ SS(X, A). Then, Int(F, A) = (RF,A, A), where the map RF,A : A → P(X) defined as follows: RF,A(p) = F(p) ∩ int(F, p), for every p ∈ A. Proof. We need to prove that (a) (RF,A, A) ⊑ (F, A), (b) (RF,A, A) ∈ τ, and (c) (G, A) ⊑ (RF,A, A), for every (G, A) ∈ τ such that (G, A) ⊑ (F, A). (a) First we observe that RF,A(p) = F(p)∩int(F, p) ⊆ F(p), for every p ∈ A. Thus, (RF,A, A) ⊑ (F, A). (b) We prove that (RF,A, A) ∈ τ. Let a ∈ A and x ∈ RF,A(a) = F(a) ∩ int(F, a). By Proposition 3.6, it suffices to prove that there exists an a-soft open neighborhood (G(a,x), A) of x such that (G(a,x), A) ⊑ (RF,A, A) or equivalently G(a,x)(p) ⊆ F(p) ∩ int(F, p), for every p ∈ A. Since x ∈ int(F, a), there exists an a-soft open neighborhood (G(a,x), A) of x such that (G(a,x), A) ⊑ (F, A). Therefore, G(a,x)(p) ⊆ F(p), for every p ∈ A. We prove that G(a,x)(p) ⊆ int(F, p), for every p ∈ A. Indeed, let y ∈ G(a,x)(p), where p ∈ A. Then, the soft set (G(a,x), A) is a p-soft open neigh- borhood of y such that (G(a,x), A) ⊑ (F, A) and, therefore, y ∈ int(F, p). Thus, G(a,x)(p) ⊆ F(p) ∩ int(F, p), for every p ∈ A. (c) Finally, let (G, A) ∈ τ such that (G, A) ⊑ (F, A). (3) We must prove that (G, A) ⊑ (RF,A, A). It suffices to prove that G(p) ⊆ F(p) ∩ int(F, p), for every p ∈ A. Indeed, let y ∈ G(p), where p ∈ A. Then, (G, A) is a p-soft open neighborhood of y such that (G, A) ⊑ (F, A). Therefore, y ∈ int(F, p). Also, by relation (3) we have G(p) ⊆ F(p). Hence, y ∈ F(p). Thus, y ∈ F(p) ∩ int(F, p). � Proposition 3.13. Let (X, τ, A) be a soft space, a ∈ A, and (F, A) ∈ SS(X, A). Then, cl((F, a)c) = X \ int(F, a). Proof. We prove that cl((F, a)c) ⊆ X \ int(F, a). Let x ∈ cl((F, a)c). Then, for every a-soft open neighborhood (G, A) of x we have (F, A)c ⊓ (G, A) 6= 0A. We suppose that x ∈ int(F, a). Then, there exists an a-soft open neighborhood c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 102 Soft set theory and topology (G, A) of x such that (G, A) ⊑ (F, A). Therefore, (F, A)c ⊓ (G, A) = 0A, which is a contradiction. Thus, x ∈ X \ int(F, a). Now, we prove that X \ int(F, a) ⊆ cl((F, a)c). Let x ∈ X \ int(F, a) and (G, A) be an a-soft open neighborhood of x. We must prove that (F, A)c ⊓ (G, A) 6= 0A. Since x /∈ int(F, a), there exists p ∈ A such that G(p) * F(p). This means that there exists x ∈ X such that x ∈ G(p) and x ∈ X \ F(p). Hence, (X \ F(p)) ∩ G(p) 6= ∅ and, therefore, (F, A)c ⊓ (G, A) 6= 0A. Thus, x ∈ cl((F, a)c). � Definition 3.14. Let (X, τ, A) be a soft topological space. The soft boundary Bd(F, A) of (F, A) ∈ SS(X, A) is the soft set Cl(F, A) ⊓ Cl((F, A)c). Proposition 3.15. Let (X, τ, A) be a soft space and (F, A) ∈ SS(X, A). Then, Bd(F, A) = (RF,A, A), where the map RF,A : A → P(X) defined as follows: RF,A(p) = (F(p) ∪ cl(F, p)) ∩ ((X \ F(p)) ∪ (X \ int(F, p))), for every p ∈ A. Proof. By Propositions 3.9, 3.12, and 3.13 for every p ∈ A we have RF,A(p) = (F(p) ∪ cl(F, p)) ∩ ((X \ F(p)) ∪ cl((F, p) c))) = (F(p) ∪ cl(F, p)) ∩ ((X \ F(p)) ∪ (X \ int(F, p))). � Definition 3.16. Let (X, τ, A) be a soft topological space. A family B ⊆ τ is called a base for (X, τ, A) if for every soft open set (G, A) 6= 0A, there exist (Gi, A) ∈ B, i ∈ I, such that (G, A) = ⊔{(Gi, A) : i ∈ I}. Proposition 3.17. Let (X, τ, A) be a soft topological space. Then, a family B ⊆ τ is a base for (X, τ, A) if and only if for every a ∈ A, x ∈ X, and every a-soft open neighborhood (G, A) of x there exists an a-soft open neighborhood (G(a,x), A) of x such that (G(a,x), A) ∈ B and (G(a,x), A) ⊑ (G, A). Proof. Let B be a base for (X, τ, A), a ∈ A, x ∈ X, and (G, A) be an a-soft open neighborhood of x. Then, x ∈ G(a). Since B is a base for (X, τ, A), there exist (Gi, A) ∈ B, i ∈ I, such that (G, A) = ⊔{(Gi, A) : i ∈ I}. Hence, G(a) = ∪{Gi(a) : i ∈ I} and, therefore, x ∈ Gi0(a) for some i0 ∈ I. Thus, (Gi0, A) is an a-soft open neighborhood of x such that (Gi0, A) ∈ B and (Gi0 , A) ⊑ (G, A). Conversely, let B ⊆ τ. Suppose that for every a ∈ A, x ∈ X, and every a-soft open neighborhood (G, A) of x there exists an a-soft open neighborhood (G(a,x), A) of x such that (G(a,x), A) ∈ B and (G(a,x), A) ⊑ (G, A). We prove that B is a base for (X, τ, A). Indeed, let (G, A) 6= 0A be a soft open set. We consider the set I = {(a, x) : a ∈ A, x ∈ G(a)}. Then, as in the proof of Proposition 3.6 we have (G, A) = ⊔{(G(a,x), A) : (a, x) ∈ I}. Since (G(a,x), A) ∈ B, for every (a, x) ∈ I, the set B is a base for (X, τ, A). � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 103 D. N. Georgiou and A. C. Megaritis Definition 3.18. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces, x ∈ X, and e : A → B. A map f : X → Y is called soft e-continuous at the point x if for every a ∈ A and every e(a)-soft open neighborhood (G, B) of f(x) in (Y, τY , B) there exists an a-soft open neighborhood (F, A) of x in (X, τX, A) such that Φfe(F, A) ⊑ (G, B). If the map f is soft e-continuous at any point x ∈ X, then we say that the map f is soft e-continuous. Proposition 3.19. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces and e : A → B. Then, the following statements are equivalent: (1) The map f : X → Y is soft e-continuous. (2) Φ−1 fe (G, B) ∈ τX, for every (G, B) ∈ τY . (3) Φ−1 fe (Q, B) ∈ τcX, for every (Q, B) ∈ τ c Y . Proof. (1) ⇒ (2) Let (G, B) ∈ τY . Then, Φ −1 fe (G, B) is the soft set (F, A) ∈ SS(X, A) such that F(pX) = f −1(G(e(pX))), for every pX ∈ A. Let a ∈ A and x ∈ F(a). By Proposition 3.6, it suffices to prove that there exists an a-soft open neighborhood (F(a,x), A) of x in (X, τX, A) such that (F(a,x), A) ⊑ (F, A). Since x ∈ F(a) = f−1(G(e(a))), we have f(x) ∈ G(e(a)). This means that the soft set (G, B) is an e(a)-soft open neighborhood of f(x) in (Y, τY , B). Since the map f : X → Y is soft e-continuous, there exists an a-soft open neighborhood (F(a,x), A) of x in (X, τX, A) such that Φfe(F(a,x), A) ⊑ (G, B). Therefore, by Proposition 2.18, (F(a,x), A) ⊑ Φ −1 fe (Φfe(F(a,x), A)) ⊑ Φ −1 fe (G, B). Thus, Φ−1 fe (G, B) ∈ τX. (2) ⇒ (1) Let x ∈ X, a ∈ A, and (G, B) be an e(a)-soft open neighborhood of f(x) in (Y, τY , B). Then, f(x) ∈ G(e(a)) or x ∈ f −1(G(e(a))). Therefore, by assumption, Φ−1 fe (G, B) is an a-soft open neighborhood of x in (X, τX, A). Therefore, by Proposition 2.18, Φfe(Φ −1 fe (G, B)) ⊑ (G, B). Thus, the map f : X → Y is soft e-continuous at the point x. (2) ⇒ (3) Let (Q, B) ∈ τcY . Then, (Q, B) c ∈ τY . By Proposition 2.18 we have Φ−1 fe ((Q, B)c) = (Φ−1 fe (Q, B))c. Since Φ−1 fe ((Q, B)c) ∈ τX, we have Φ−1 fe (Q, B) ∈ τcX. (3) ⇒ (2) Let (G, B) ∈ τY . Then, (G, B) c ∈ τcY . By Proposition 2.18 we have Φ−1 fe ((G, B)c) = (Φ−1 fe (G, B))c. Since Φ−1 fe ((G, B)c) ∈ τcX, we have Φ−1 fe (G, B) ∈ τX. � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 104 Soft set theory and topology Example 3.20. Let X = {x1, x2, x3}, Y = {y1, y2, y3}, A = {0, 1}, and B = {0, 1, 2}. We consider the following soft sets (F, A), (G, A), and (H, A) over X defined as follows: F(p) = { {x3}, if p = 0, {x1, x2}, if p = 1, G(p) = { ∅, if p = 0, {x3}, if p = 1, H(p) = { {x3}, if p = 0, X, if p = 1. Also, we consider the following soft sets (Q, B) and (R, B) over Y defined as follows: Q(p) =      {y1}, if p = 0, {y3}, if p = 1, ∅, if p = 2, R(p) =      {y1, y2}, if p = 0, {y3}, if p = 1, Y, if p = 2. Then, the triplets (X, τX, A) and (Y, τX, B), where τX = {0A, 1A, (F, A), (G, A), (H, A)} and τY = {0B, 1B, (Q, B), (R, B)} are soft topological spaces. Let f : X → Y be the map such that f(x1) = f(x2) = y1 and f(x3) = y3 and e : A → B the map such that e(0) = 1 and e(1) = 0. Then, the map f is soft e-continuous. Also, if e′ : A → B is the map such that e′(0) = 1 and e′(1) = 2, then the map f is not soft e′-continuous. Proposition 3.21. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces, BY a base for (Y, τY , B), and e : A → B. Then, the following statements are equivalent: (1) The map f : X → Y is soft e-continuous. (2) Φ−1 fe (G, B) ∈ τX, for every (G, B) ∈ BY . Proof. (1) ⇒ (2) Follows by Proposition 3.19. (2) ⇒ (1) By Proposition 3.19 it suffices to prove that Φ−1 fe (G, B) ∈ τX, for every (G, B) ∈ τY . Let (G, B) ∈ τY . Then, there exist (Gi, B) ∈ BY , i ∈ I, such that (G, B) = ⊔{(Gi, B) : i ∈ I}. Therefore, by Proposition 2.19 we have Φ−1 fe (G, B) = Φ−1 fe (⊔{(Gi, B) : i ∈ I}) = ⊔{Φ −1 fe (Gi, B) : i ∈ I} ∈ τX. � Remark 3.22 (see, for example, [24]). Let (X, τ, A) be a soft topological space and (F, A) ∈ SS(X, A). We recall the following properties : (1) (F, A) ∈ τc if and only if Cl(F, A) = (F, A). (2) (F, A) ∈ τ if and only if Int(F, A) = (F, A). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 105 D. N. Georgiou and A. C. Megaritis (3) Int(F, A) = (Cl((F, A)c))c. (4) Cl(F, A) = (Int((F, A)c))c. (5) If (F, A) ⊑ (G, A), then Cl(F, A) ⊑ Cl(G, A). Proposition 3.23. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces and e : A → B. Then, the following statements are equivalent: (1) The map f : X → Y is soft e-continuous. (2) Φfe(Cl(F, A)) ⊑ Cl(Φfe(F, A)), for every (F, A) ∈ SS(X, A). (3) Cl(Φ−1 fe (G, B)) ⊑ Φ−1 fe (Cl(G, B)), for every (G, B) ∈ SS(Y, B). (4) Φ−1 fe (Int(G, B)) ⊑ Int(Φ−1 fe (G, B)), for every (G, B) ∈ SS(Y, B). Proof. (1) ⇒ (2) Let (F, A) ∈ SS(X, A). Since Φfe(F, A) ⊑ Cl(Φfe(F, A)), by Proposition 2.18 we have (F, A) ⊑ Φ−1 fe (Φfe(F, A)) ⊑ Φ −1 fe (Cl(Φfe(F, A))). Therefore, Cl(F, A) ⊑ Cl(Φ−1 fe (Cl(Φfe(F, A)))). Since Cl(Φfe(F, A)) ∈ τ c Y , by Proposition 3.19, Φ −1 fe (Cl(Φfe(F, A))) ∈ τ c X and, therefore, Cl(Φ−1 fe (Cl(Φfe(F, A)))) = Φ −1 fe (Cl(Φfe(F, A))). Hence, Cl(F, A) ⊑ Φ−1 fe (Cl(Φfe(F, A))). Finally, by Proposition 2.18 we have Φfe(Cl(F, A)) ⊑ Φfe(Φ −1 fe (Cl(Φfe(F, A)))) ⊑ Cl(Φfe(F, A)). (2) ⇒ (3) Let (G, B) ∈ SS(Y, B). We apply (2) to (F, A) = Φ−1 fe (G, B) and we obtain the inclusion Φfe(Cl(Φ −1 fe (G, B))) ⊑ Cl(Φfe(Φ −1 fe (G, B))) ⊑ Cl(G, B). Therefore, Cl(Φ−1 fe (G, B)) ⊑ Φ−1 fe (Φfe(Cl(Φ −1 fe (G, B)))) ⊑ Φ−1 fe (Cl(G, B)). (3) ⇒ (4) Let (G, B) ∈ SS(Y, B). We apply (3) to (G, B)c and we obtain the inclusion Cl(Φ−1 fe ((G, B)c)) ⊑ Φ−1 fe (Cl((G, B)c)), which gives (see Proposition 2.18) Φ−1 fe (Int(G, B)) = Φ−1 fe ((Cl((G, B)c))c) = (Φ−1 fe (Cl((G, B)c)))c ⊑ (Cl(Φ−1 fe ((G, B)c)))c = (Cl((Φ−1 fe (G, B))c))c = Int(Φ−1 fe (G, B)). (4) ⇒ (1) By Proposition 3.19 it suffices to prove that Φ−1 fe (G, B) ∈ τX, for every (G, B) ∈ τY . Let (G, B) ∈ τY . Then, Int(G, B) = (G, B). Therefore, Φ−1 fe (G, B) = Φ−1 fe (Int(G, B)) ⊑ Int(Φ−1 fe (G, B)). c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 106 Soft set theory and topology Also, Int(Φ−1 fe (G, B)) ⊑ Φ−1 fe (G, B). Thus, Int(Φ−1 fe (G, B)) = Φ−1 fe (G, B), which means that Φ−1 fe (G, B) ∈ τX. � Definition 3.24. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces and e : A → B. A map f : X → Y is called soft e-open (respectively, soft e- closed) if for every (F, A) ∈ τX (respectively, (F, A) ∈ τ c X) we have Φfe(F, A) ∈ τY (respectively, Φfe(F, A) ∈ τ c Y ). Proposition 3.25. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces and e : A → B. Then, the following statements are equivalent: (1) The map f : X → Y is soft e-open. (2) Φfe(Int(F, A)) ⊑ Int(Φfe(F, A)), for every (F, A) ∈ SS(X, A). Proof. (1) ⇒ (2) Let (F, A) ∈ SS(X, A). Since, Int(F, A) ⊑ (F, A), we have Φfe(Int(F, A)) ⊑ Φfe(F, A). Since Int(F, A) ∈ τX, we have Φfe(Int(F, A)) ∈ τY . Therefore, by the above inclusion we have Φfe(Int(F, A)) ⊑ ⊔{(G, A) ∈ τY : (G, A) ⊑ Φfe(F, A)} = Int(Φfe(F, A)). (2) ⇒ (1) We prove that Φfe(F, A) ∈ τY , for every (F, A) ∈ τX. Let (F, A) ∈ τX. Then, Φfe(F, A) = Φfe(Int(F, A)) ⊑ Int(Φfe(F, A)). Also, Int(Φfe(F, A)) ⊑ Φfe(F, A). Thus, Int(Φfe(F, A)) = Φfe(F, A), which means that Φfe(F, A) ∈ τY . � The proof of the following proposition is similar to the proof of Proposition 3.25. Proposition 3.26. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces and e : A → B. Then, the following statements are equivalent: (1) The map f : X → Y is soft e-closed. (2) Cl(Φfe(F, A)) ⊑ Φfe(Cl(F, A)), for every (F, A) ∈ SS(X, A). Definition 3.27. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces and e a 1-1 map of A onto B. A soft e-continuous map f of X onto Y is called soft e-homeomorphism if the map f is 1-1 and the inverse map f−1 : Y → X is soft e−1-continuous. Proposition 3.28. Let (F, A) ∈ SS(X, A), e a 1-1 map of A onto B, and f a 1-1 map of X onto Y . Then, (1) Φfe(F, A) = Φ −1 f−1e−1 (F, A). (2) Φfe((F, A) c) = (Φfe(F, A)) c. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 107 D. N. Georgiou and A. C. Megaritis Proof. (1) Let Φfe(F, A) = (G, B), Φ −1 f−1e−1 (F, A) = (G′, B), and pY ∈ B. We must prove that G(pY ) = G ′(pY ). Let e −1(pY ) = pX. Since the map e : A → B is 1-1, G(pY ) = f(F(pX)). On the other hand, G′(pY ) = (f −1)−1(F(e−1(pY ))) = f(F(pX)). Thus, G(pY ) = G ′(pY ). (2) Let Φfe((F, A) c) = (G, B), (Φfe(F, A)) c = (G′, B), and pY ∈ B. We must prove that G(pY ) = G ′(pY ). Let e −1(pY ) = pX. Since the map e : A → B is 1-1, G(pY ) = f(X \ F(pX)). Since the map f : X → Y is 1-1 and onto, f(X \ F(pX)) = Y \ f(F(pX)). Therefore, G(pY ) = Y \ f(F(pX)). On the other hand, G′(pY ) = Y \ f(F(pX)). Thus, G(pY ) = G ′(pY ). � Proposition 3.29. Let (X, τX, A) and (Y, τY , B) be two soft topological spaces, e a map of A onto B, and f a 1-1 map of X onto Y . Then, the following statements are equivalent: (1) The map f is soft e-homeomorphism. (2) The map f is soft e-continuous and soft e-open. (3) The map f is soft e-continuous and soft e-closed. Proof. (1) ⇒ (2) We prove that Φfe(F, A) ∈ τY , for every (F, A) ∈ τX. Let (F, A) ∈ τX. Since the map f −1 is soft e−1-continuous and (F, A) ∈ τX, we have Φ−1 f−1e−1 (F, A) ∈ τY . By Proposition 3.28, Φfe(F, A) ∈ τY . (2) ⇒ (3) We prove that Φfe(F, A) ∈ τ c Y , for every (F, A) ∈ τ c X. Let (F, A) ∈ τcX. Then, (F, A) c ∈ τX and, therefore, Φfe((F, A) c) ∈ τY . By Proposition 3.28, (Φfe(F, A)) c = Φfe((F, A) c), which means that Φfe(F, A) ∈ τ c Y . (3) ⇒ (1) We prove that the inverse map f−1 : Y → X is soft e−1- continuous. It suffices to prove that Φ−1 f−1e−1 (F, A) ∈ τcY , for every (F, A) ∈ τ c X. Let (F, A) ∈ τcX. Then, Φfe(F, A) ∈ τ c Y . 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