On Pairwise Soft Connectedness

Math. Sci. Lett. 7, No. 2, 79-89 (2018)

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Mathematical Sciences Letters An International Journal http://dx.doi.org/10.18576/msl/070202

On Pairwise Soft Connectedness Ali Kandil1 , Osama El Tantawy2 , Sobhy El-Sheikh3 and Shawqi Ahmed Hazza4,∗ 1 Department

of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt 3 Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt 4 Department of Mathematics, Faculty of Education, Taiz University, Taiz, Yemen 2 Department

Received: 4 Jun. 2017, Revised: 2 Apr. 2018, Accepted: 13 Apr. 2018 Published online: 1 May 2018

Abstract: The purpose of this paper is to use the concept of soft ideal bitopological space (X, η1 , η2 , E, I˜) to introduce the pairwise ˜

˜

soft local function denoted by ()∗12I . The properties of pairwise soft local function ()∗12I and some important results related to it have ˜

∗I to generate a family η ∗ which is finer than η and we show that η ∗ is a supra obtained. We using the pairwise soft local function ()12 12 12 12 ∗ is used to study the pairwise soft connectedness soft topology but not a soft topology in general. In addition, the supra soft topology η12

in soft ideal bitopological space (X, η1 , η2 , E, I˜). Keywords: Soft set; Soft topology; Soft ideal; Soft local function; Pairwise soft local function; Soft bitopological spaces; Pairwise soft connectedness; p∗ -soft connected spaces, p∗ -connected soft set; p∗ -soft continuous. Mathematics Subject Classification (2010): 06D72, 54A40, 54E55, 54D10.

1 Introduction In many entangled issues emerging in the field of designing, sociology, financial matters, and therapeutic science. This includes instabilities, established strategies which are found to be used in this article. Molodtsov [20] pointed out that the important theories such as: Probability Theory and Fuzzy set Theory. which can be considered as a mathematical tools for dealing with uncertainties have their own troubles. In addition, the author called attention to that the explanation behind these challenges is, potentially, the insufficiency of the parameterizations instrument of the hypothesis. Further-more, the author started the novel idea of soft set as another scientific instrument for managing instabilities. Soft set theory is free of the troubles display in these theories. This lauds that the soft systems give an extremely broad structure with the contribution of parameters. Therefore, researches work on soft set theory and its applications in various fields are progressing rabidly. Pei and Miao in [25] discussed the relationship between soft sets and information systems. Hussain et. al. [7] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of ∗ Corresponding

parameters. They also studied some of basic concepts of soft topological spaces. The notion of soft ideal is initiated for the first time by Kandil et al. [10]. They also introduced the concept of soft local function.

In 2014, M. Al-Khafaj and M. Mahmood [1] introduced and studied some properties of soft connected spaces and soft locally connected spaces. Ittanagi [9] introduced the concept of soft bitopological space and studied some types of soft separation axiom for soft bitopological spaces from his point of view. Kandil et al. in [[13],[14],[16], [17]] introduced some structures of soft bitopological spaces. They introduced some types of pairwise soft open (continuous) mappings and some related results. Recently, Kandil et al. [12] introduced and studied the notions of pairwise soft connected in soft bitopological spaces.

In this paper, given a soft bitopological space (X, η1 , η2 , E) and its associated supra soft topological ˜ E), space (X, η12 , E). Also, let I˜ be a soft ideal on (X, we introduce the pairwise soft local function,

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Natural Sciences Publishing Cor.

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A. Kandil et al.: On pairwise soft connectedness

˜

()∗12I : SS(X)E → SS(X)E such that ˜ ˜ E) 6∈ (G, E)∗12I = {xe ∈ ξ (X)E : (G, E)∩(O, I˜ ∀ (O, E) ∈ η12 (xe )} where η12 = {(G, E) ∈ SS(X)E : ˜ 2 , E), (Gi , E) ∈ η12 , i = 1, 2}. The (G, E) = (G1 , E)∪(G properties of the operator ()∗12 have obtained. In addition, ˜ we used the pairwise soft local function ()∗12I to generate ∗ ∗ a family η12 which is finer than η12 and we show that η12 is a supra soft topology but not a soft topology in general. ∗ is used to study Furthermore, the supra soft topology η12 the pairwise soft connectedness in soft ideal bitopological space (X, η1 , η2 , E, I˜).

2 Preliminaries In this section, we briefly review some concepts and some related results of soft set, soft topological space and soft bitopological space which are needed to used in current paper. For more details about these concepts you can see [[3], [4], [5], [6], [7], [8], [9], [11], [14], [15], [20], [21], [22], [25], [26], [27]]. Let X be an initial universe, E be a set of parameters and P(X) be the power set of X. Definition 2.1.[22] A pair (F, E) is called a soft set over X, where F is a mapping given by F : E → P(X). A soft set can also be defined by the set of ordered pairs (F, E) = {(e, F(e)) : e ∈ E, F : E → P(X)}. From now on, SS(X)E denotes the family of all soft sets over X with a fixed set of parameters E. For two soft sets (F, E), (G, E) ∈ SS(X)E , (F, E) is called ˜ a soft subset of (G, E), denoted by (F, E)⊆(G, E), if F(e) ⊆ G(e), ∀e ∈ E. In this case, (G, E) is called a soft superset of (F, E). In addition, the union of soft sets ˜ (F, E) and (G, E), denoted by (F, E)∪(G, E), is the soft set (H, E) which defined as H(e) = F(e) ∪ G(e), ∀e ∈ E. Moreover, the intersection of soft sets (F, E) and (G, E), ˜ denoted by (F, E)∩(G, E), is the soft set (M, E) which defined as, M(e) = F(e) ∩ G(e), ∀e ∈ E. The complement of a soft set (F, E), denoted by (F, E)c , is defined as, (F, E)c = (F c , E), where F c : E → P(X) is a mapping given by F c (e) = X \ F(e), ∀ e ∈ E. The difference of soft sets (F, E) and (G, E), denoted by (F, E) \ (G, E), is the soft set (H, E), which defined as H(e) = F(e) \ G(e), ˜ ∀ e ∈ E. Clearly, (F, E) \ (G, E) = (F, E)∩(G, E)c . A soft set (F, E) is called a null soft set, denoted by (φ˜ , E) , if F(e) = φ , ∀ e ∈ E. Moreover, a soft set (F, E) is called an ˜ E), if F(e) = X, ∀e ∈ E. absolute soft set, denoted by (X, ˜ E) and (X, ˜ E)c = (φ˜ , E). Clearly, we have (φ˜ , E)c = (X, Moreover, a soft set (G, E) is said to be a finite soft set if G(e) is a finite set for all e ∈ E. Otherwise, it is called an infinite soft set. A soft set (G, E) over X is called a proper ˜ E) if (G, E) ( (X, ˜ E). soft subset of (X, Definition 2.2.[[2],[3], [21],[23]] A soft set ˜ E) if there (F, E) ∈ SS(X)E is said to be a soft point in (X,

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exist x ∈ X and e ∈ E such that F(e) = {x} and F(e′ ) = φ for each e′ ∈ E \ {e}. This soft point is denoted by (xe , E) or xe , i.e., xe : E → P(X) is a mapping defined as {x} i f e = a, xe (a) = for all a ∈ E. φ i f e 6= a A soft point (xe , E) is said to be belonging to the soft ˜ E), if xe (e) ⊆ G(e), i.e., set (G, E), denoted by xe ∈(G, ˜ E) if and only if {x} ⊆ G(e). Clearly, xe ∈(G, ˜ (xe , E)⊆(G, E). In addition, two soft points xe1 , ye2 over X are said to be equal if x = y and e1 = e2 . Thus, xe1 6= ye2 iff x 6= y or e1 6= e2 . ˜ E) is denoted by The family of all soft points in (X, ξ (X)E . Proposition 2.1.[3] The union of any collection of soft points can be considered as a soft set and every soft set can be expressedSas a union of all soft points belonging ˜ to it, i.e., (G, E) = ˜ {(xe , E) : xe ∈(G, E)}. Proposition 2.2.[3] Let (G, E), (H, E) be two soft sets over X. Then, ˜ E) ⇔ xe 6∈˜ (G, E)c . (1)xe ∈(G, ˜ ˜ ˜ ˜ E). E) or xe ∈(H, E)∪(H, E) ⇔ xe ∈(G, (2)xe ∈(G, ˜ ˜ ˜ ˜ E). E) and xe ∈(H, E)∩(H, E) ⇔ xe ∈(G, (3)xe ∈(G, ˜ ˜ ˜ E)]. E) ⇒ xe ∈(H, (4)(G, E)⊆(H, E) ⇔ [xe ∈(G, For more details for soft point you can see in [[3], [21],[23]]. Definition 2.3.[[24],[26]] Let η be a collection of soft sets over a universe X with a fixed set of parameters E, i.e., η ⊆ SS(X)E . The collection η is called a soft topology on X if it satisfies the following axioms: ˜ E), (φ˜ , E) ∈ η , ‘ (1)(X, (2)The union of any number of soft sets in η belongs to η , (3)The intersection of any two soft sets in η belongs to η . The triple (X, η , E) is called a soft topological space. Any member of η is said to be an open soft set in (X, η , E). A soft set (F, E) over X is said to be a closed soft set in (X, η , E), if its complement (F, E)c is an open soft set in (X, η , E). We denote the family of all closed soft sets by η c . Definition 2.4.[26] Let (X, η , E) be a soft topological space and (F, E) ∈ SS(X)E . The soft closure of (F, E), denoted by sclη (F, E)(or scl(F, E)), is defined by T ˜ sclη (F, E) = ˜ {(H, E) ∈ η c : (F, E)⊆(H, E)}).

Definition 2.5.[1] The two soft sets (A, E), (B, E) in a soft topological space (X, η , E) are called separated soft sets if ˜ ˜ E) = sclη (B, E)∩(A, E) = (φ˜ , E). sclη (A, E)∩(B, Definition 2.6.[1] A soft topological space (X, η , E) is called soft connected space if X cannot be expressed as a union of two separated soft sets. Definition 2.7.[1] A soft set (G, E) in a soft topological space (X, η , E) is called a disconnected soft set if there exist two separated soft sets (A, E), (B, E) such that

Math. Sci. Lett. 7, No. 2, 79-89 (2018) / www.naturalspublishing.com/Journals.asp

˜ (G, E) = (A, E)∪(B, E). Otherwise, (G, E) is called a connected soft set. Definition 2.8.[18],[27] Let SS(X)E and SS(Y )K be two families of soft sets. Let u : X → Y and p : E → K be mappings. We define a soft mapping f pu : SS(X)E → SS(Y )K as: (i)If (G, E) ∈ SS(X)E . Then, the image of (G, E) under f pu , written as f pu (G, E) = ( f pu (G), p(E)), is a soft set in SS(Y )K such that S −1 e∈p−1 (k) u[G(e)] i f p (k) 6= φ , f pu (G, E)(k) = −1 φ i f p (k) = φ for all k ∈ K. (ii)If (H, K) ∈ SS(Y )K . Then, the inverse image of (H, K) −1 (H, K) = ( f −1 (H), E), is a under f pu , written as f pu pu soft set in SS(X)E such that −1 f pu (H, K)(e) = u−1 [H(p(e))] ∀ e ∈ E.

Theorem 2.1.[18] Let SS(X)E and SS(Y )K be two families of soft sets. For a soft mapping f pu : SS(X)E → SS(Y )K we have the following: ˜ E)⊆( ˜ Y˜ , K). (1) f pu (φ˜ , E) = (φ˜ , K) and f pu (X, S S ˜ i∈I f pu (Gi , E) (2) f pu [ ˜ i∈I (Gi , E)] = and T T ˜ ˜ ˜ f pu [ i∈I (Gi , E)]⊆ i∈I f pu (Gi , E). ˜ ˜ f pu (M, E). (3)(G, E)⊆(M, E) ⇒ f pu (G, E)⊆ Theorem 2.2.[18],[19] Let SS(X)E and SS(Y )K be two families of soft sets. For a soft mapping f pu : SS(X)E → SS(Y )K we have the following: −1 (φ˜ , K) = (φ˜ , E) and f −1 (Y˜ , K) = (X, ˜ E). (1) f pu pu S S −1 −1 ˜ ˜ (2) f pu [ i∈I (Hi , K)] = i∈I f pu (Hi , K) −1 [T ˜ i∈I (Hi , K)] = T ˜ i∈I f −1 (Hi , K). f pu pu −1 (H, K)⊆ −1 (N, K). ˜ ˜ f pu (3)(H, K)⊆(N, K) ⇒ f pu

and

Theorem 2.3.[27] Let SS(X)E and SS(Y )K be two families of soft sets. For a soft mapping f pu : SS(X)E → SS(Y )K we have the following: −1 [(Y˜ , K) \ (H, K)] = (X, −1 (H, K)] for any ˜ E) \ [ f pu (1) f pu (H, K) ∈ SS(Y )K . −1 [ f (G, E)] for any (G, E) ∈ SS(X) . If f ˜ f pu (2)(G, E)⊆ pu E pu is a soft injective, the equality holds. −1 (H, K)]⊆(H, ˜ (3) f pu [ f pu K) for any (H, K) ∈ SS(Y )K . If f pu is a soft surjective, the equality holds. Definition 2.9.[9] A quadrable system (X, η1 , η2 , E) is called a soft bitopological space [briefly, soft bitopological space], where η1 , η2 are arbitrary soft topologies on X with a fixed set of parameters E. Definition 2.10.[14] Let (X, η1 , η2 , E) be a soft bitopological space. A soft set (G, E) over X is said to be a pairwise open soft set in (X, η1 , η2 , E) [briefly, p-open soft set] if there exist an open soft set (G1 , E) in η1 and an open soft set (G2 , E) in η2 such that

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˜ 2 , E). A soft set (G, E) over X is said (G, E) = (G1 , E)∪(G to be a pairwise closed soft set in (X, η1 , η2 , E) [briefly, p-closed soft set] if its complement is a p-open soft set in (X, η1 , η2 , E). Clearly, a soft set (F, E) over X is a p-closed soft set in (X, η1 , η2 , E) if there exist an η1 -closed soft set (F1 , E) and an η2 -closed soft set (F2 , E) ˜ 2 , E). such that (F, E) = (F1 , E)∩(F The family of all p-open (p-closed) soft sets in a soft c ), bitopological space (X, η1 , η2 , E) is denoted by η12 (η12 respectively. Theorem 2.4.[14] Let (X, η1 , η2 , E) be a soft bitopological space. The family of all p-open soft sets η12 is a supra soft topology on X, where ˜ 2 , E) : (Gi , E) ∈ ηi , i = 1, 2}. η12 = {(G, E) = (G1 , E)∪(G The triple (X, η12 , E) is the supra soft topological space associated to the soft bitopological space (X, η1 , η2 , E). Lemma 2.1.[11] For any soft bitopological space (X, η1 , η2 , E), we have (X, η1e , η2e ) as an ordinary bitopological space, for all e ∈ E,where ηie = {G(e) : (G, E) ∈ ηi }, i = 1, 2, and e = η (e) = {H(e) : (H, E) ∈ η }. η12 12 12 Definition 2.11.[14] Let (X, η1 , η2 , E) be a soft bitopological space and let (G, E) ∈ SS(X)E . The pairwise soft closure of (G, E), denoted by sclη12 (G, E)[or scl12 (G, E) ], is defined by T c : (G, E)⊆(F, ˜ scl12 (G, E) = ˜ {(F, E) ∈ η12 E)}. Corollary 2.1.[14] Let (X, η1 , η2 , E) be a soft bitopological space and (G, E) ∈ SS(X)E . Then, ˜ η2 (G, E). sclη12 (G, E) = sclη1 (G, E)∩scl Theorem 2.5.[14] Let (X, η1 , η2 , E) be a bitopological space and (G, E) ∈ SS(X)E . Then,

soft

˜ 12 (G, E) ⇔ (Oxe , E)∩(G, ˜ E) 6= xe ∈scl (φ˜ , E), ∀ (Oxe , E) ∈ η12 (xe ), where (Oxe , E) is any p-open soft set contains xe and η12 (xe ) is the family of all p-open soft sets which containing xe . For more details about the properties of pairwise soft closure operator see in [14]. Theorem 2.6.[12] Let (X, η1 , η2 , E) be a soft bitopological space. Then the following are equivalent: (1)(X, η1 , η2 , E) is p-soft connected. ˜ E) cannot represented as a union of two non-null (2)(X, disjoint p-open soft sets. ˜ E) cannot represented as a union of two non null (3)(X, disjoint p-closed soft sets. ˜ E) has no proper soft subset which is both p-open (4)(X, and p-closed soft set other than (φ˜ , E). Definition 2.12.[10] A nonempty collection I˜ of soft sets over a universe X with the same set of parameters E is ˜ E) if called a soft ideal on (X, ˜ (1)(F, E) ∈ I˜ and (G, E)⊆(F, E) ⇒ (G, E) ∈ I˜,

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˜ (2)(F, E) ∈ I˜ and (G, E) ∈ I˜ ⇒ (F, E)∪(G, E) ∈ I˜.

(G5 , E) = {(e1 , {y}), (e2 , {x, y})},

Definition 2.13.[10] Let (X, η , E) be a soft topological ˜ E). The operator space and I˜ be a soft ideal on (X, ˜ ˜ ∗ I ()η : SS(X)E → SS(X)E given by (F, E)∗ηI = {xe ∈ ˜ E) 6∈ I˜ ∀ (Oxe , E) ∈ η } is called a ξ (X)E : (Oxe , E)∩(F, soft local function. When there is no ambiguity we will ˜ write (F, E)∗ instead of (F, E)η∗I . Theorem 2.7.[10] Let (X, η , E) be a soft topological space ˜ E). Then, and I˜, J˜ be two soft ideals on (X,

Theorem 2.8.[10] Let (X, η , E) be a soft topological space ˜ E). Then, the operator sclη∗ : and I˜ be a soft ideal on (X, SS(X)E → SS(X)E defined by: ˜ E)η∗ sclη∗ (F, E) = (F, E)∪(F, is a soft closure operator and induced the unique soft topology

η ∗ = {(F, E) ∈ SS(X)E : sclη∗ (F, E)c = (F, E)c } which is a finer than η .

3 Pairwise soft local function Definition 3.1. Let (X, η1 , η2 , E) be a soft bitopological ˜ E). The system space, I˜ be a soft ideal on (X, ˜ (X, η1 , η2 , E, I ) is called a soft ideal bitopological space. Definition 3.2. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space and let (G, E) ∈ SS(X)E . The ˜ operator ()∗12I : SS(X)E → SS(X)E , given by S ∗I˜ = ˜ {x ˜ E) 6∈ (G, E)12 e ∈ ξ (X)E : (Oxe , E)∩(G, I˜ ∀ (Oxe , E) ∈ η12 (xe )}, is called a pairwise soft local function associated with I˜. When there is no ambiguity ˜ we will write (G, E)∗12 instead of (G, E)∗12I . Example 3.1. Let X = {x, y, z}, E = {e1 , e2 } and let

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and (K, E) = {(e1 , {x}), (e2 , {z})}, (M, E) = {(e1 , {x}), (e2 , φ )}, (N, E) = {(e1 , φ ), (e2 , {z})}. It is easy to verify that (X, η1 , η2 , E, I˜) is a soft ideal bitopological space.

(1)(φ˜ , E)η∗ = (φ˜ , E), ˜ ˜ E)η∗ , (2)(F, E)⊆(G, E) ⇒ (F, E)∗η ⊆(G, ˜ ∗J˜ (3)I˜ ⊆ J˜ ⇒ (F, E)∗ηI ⊆ (F, E)η , ˜ ˜ η (F, E). E)η∗ = sclη (F, E)∗η ⊆scl (4)[(F, E)∗η ]∗η ⊆(F,

η1 ˜ E), (G1 , E), (G2 , E), (G3 , E), (G4 , E), {(φ˜ , E), (X, (G5 , E)}, ˜ E), (H1 , E), (H2 , E), (H3 , E)}, η2 = {(φ˜ , E), (X, I˜ = {(φ˜ , E), (K, E), (M, E), (N, E)}, where (G1 , E) = {(e1 , {y}), (e2 , {x})}, (G2 , E) = {(e1 , {y, z}), (e2 , {x, y})}, (G3 , E) = {(e1 , {x, y}), (e2 , X)}, (G4 , E) = {(e1 , {x, y}), (e2 , {x, z})},

(H1 , E) = {(e1 , {x, y}), (e2 , {x, z})}, (H2 , E) = {(e1 , {y}), (e2 , {z})}, (H3 , E) = {(e1 , {x, y}), (e2 , X)},

=

Moreover η12 ˜ E), (G1 , E), (G2 , E), (G3 , E), (G4 , E), {(φ˜ , E), (X, (G5 , E), (H1 , E), (H2 , E), (H3 , E), (P1 , E), (P2 , E), (P3 , E)}, where (P1 , E) = {(e1 , {y}), (e2 , {x, z})}, (P2 , E) = {(e1 , {y, z}), (e2 , X)}, (P3 , E) = {(e1 , {y}), (e2 , X)}.

=

Now, let (G, E) = {(e1 , {x, z}), (e2 , {z})}. Then, ˜ (G, E)∗I˜ ˜ 4 , E) but because xe1 ∈(G xe1 6∈ 12 ˜ (G4 , E)∩(G, E) = (K, E) ∈ I˜, ˜ (G, E)∗I˜ ˜ 1 , E) but because xe2 ∈(G xe2 6∈ 12 ˜ ˜ ˜ (G1 , E)∩(G, E) = (φ , E) ∈ I , ˜ (G, E)∗I˜ ˜ 1 , E) but because ye1 ∈(H ye1 6∈ 12 ˜ (H1 , E)∩(G, E) = (K, E) ∈ I˜, ˜ (G, E)∗I˜ ˜ 3 , E) but because ye2 ∈(P ye2 6∈ 12 ˜ (P3 , E)∩(G, E) = (N, E) ∈ I˜, ˜ ˜ 2 , E) but because ze2 ∈(H ze2 6∈˜ (G, E)∗12I ˜ ˜ (P3 , E)∩(G, E) = (N, E) ∈ I . ˜ ˜ E)∗12I because But ze1 ∈(G, ˜ E), (G2 , E), (P2 , E)}, and we have η12 (ze1 ) = {(X, ˜ E)∩(G, ˜ (X, E) = (G, E) 6∈ I˜, ˜ (G2 , E)∩(G, E) = {(e1 , {z}), (e2 , φ )} 6∈ I˜, ˜ (P2 , E)∩(G, E) = {(e1 , {z}), (e2 , {z})} 6∈ I˜. ˜

Consequently, (G, E)∗12I = {(e1 , {z}), (e2 , φ )}. Theorem 3.1. Let (X, η1 , η2 , E) be a soft bitopological ˜ E). Then, space and let I˜, J˜ be two soft ideals on (X, ˜ ˜ ˜ (1)(G, E)⊆(H, E) ⇒ (G, E)∗12I ⊆ (H, E)∗12I , ∀(G, E), (H, E) ∈ SS(X)E . ˜ (2)If (G, E) ∈ I˜, then (G, E)∗12I = (φ˜ , E). ∗J˜ ˜˜ (3)I˜ ⊆ J˜ ⇒ (G, E)∗12I ⊆ (G, E)12 , ∀(G, E) ∈ SS(X)E .

Proof. Straightforward. Theorem 3.2. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then,


˜ ˜ ˜ (1)(G, E)∗12 ∪(H, E)∗12 ⊆[(G, E)∪(H, E)]∗12 , for all (G, E), (H, E) ∈ SS(X)E . ˜ ˜ η12 (G, E). (2)[(G, E)∗12 ]∗12 ⊆(G, E)∗12 = sclη12 [(G, E)∗12 ]⊆scl Proof. (1): For any two soft sets (G, E), (H, E) we ˜ ˜ have (G, E), (H, E)⊆(G, E)∪(H, E). It follows by ˜ ˜ E)∪(H, E)]∗12 and Theorem 3 that (G, E)∗12 ⊆[(G, ∗ ∗ ˜ ˜ (H, E)12 ⊆[(G, E) ∪(H, E)]12 . Therefore, ˜ ˜ ˜ E)∪(H, E)]∗12 . E)∗12 ⊆[(G, (G, E)∗12 ∪(H, ˜ E)∗12 ]∗12 . Then (2): Let xe ∈[(G, ∗ ˜ (Oxe , E)∩(G, E)12 6∈˜ I˜, ∀(Oxe , E) ∈ η12 (xe ). It follows ˜ that (Oxe , E)∩(G, E)∗12 6= (φ˜ , E), ∀(Oxe , E) ∈ η12 (xe ). Therefore, for any (Oxe , E) ∈ η12 (xe ) there exists ˜ xe , E) and yα ∈(G, ˜ yα ∈ ξ (X)E such that yα ∈(O E)∗12 . ˜ E) 6∈ I˜ for all (Oyα , E) ∈ η12 (yα ). Thus, (Oyα , E)∩(G, ˜ xe , E), then (Oxe , E)∩(G, ˜ Since yα ∈(O E) 6∈ I˜, ˜ E)∗12 . Hence, ∀(Oxe , E) ∈ η12 (xe ). It follows that xe ∈(G, ˜ ˜ η12 [(G, E)∗12 ]. E)∗12 . Now, let xe ∈scl [(G, E)∗12 ]∗12 ⊆(G, Then, for every (Oxe , E) ∈ η12 (xe ) we have ˜ (Oxe , E)∩(G, E)∗12 6= (φ˜ , E). Therefore, there exists ˜ xe , E)∩(G, ˜ ˜ xe , E) E)∗12 implies yα ∈(O and yα ∈(O ∗ yα ∈ (G, E)12 . So, for every (Oyα , E) ∈ η12 (yα ) we find ˜ ˜ that (Oyα , E)∩(G, E) 6∈ I˜. Hence, (Oxe , E)∩(G, E) 6∈ I˜ ˜ E)∗12 . Therefore, which implies that xe ∈(G, ˜ sclη12 [(G, E)∗12 ]⊆(G, E)∗12 . But, ∗ ˜ then (G, E)12 ⊆sclη12 [(G, E)∗12 ], (G, E)∗12 = sclη12 [(G, E)∗12 ]. Finally, let xe 6∈˜ sclη12 (G, E). Then there exists (Oxe , E) ∈ η12 (xe ) such that ˜ (Oxe , E)∩(G, E) = (φ˜ , E) ∈ I˜. Thus, xe 6∈˜ (G, E)∗12 . Hence, (G, E)∗12 ⊆ sclη12 (G, E). This complete the proof. Proposition 3.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, (X, η1∗ , η2∗ , E) is a soft bitopological space induced by I˜. Proof. It is immediate from Theorem 2. Lemma 3.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Let ()∗12 : SS(X)E → SS(X)E be a pairwise soft local function. Then, ˜ (G, E)∗12 = (G, E)∗1 ∩(G, E)∗2 ∀ (G, E) ∈ SS(X)E , S ˜ E) 6∈ where (G, E)∗i = ˜ {xe ∈ ξ (X)E : (Oxe , E)∩(G, ˜ I ∀ (Oxe , E) ∈ ηi (xe )}, i = 1, 2.

˜ xe 6∈˜ (G, E)∗1 ∩(G, E)∗2 . Then, If xe 6∈˜ (G, E)∗1 , then

˜ (G, E)∗ xe 6∈ 1

Proof. Let or xe 6∈˜ (G, E)∗2 . there exists ˜ E) ∈ I˜. (Oxe , E) ∈ η1 ⊆ η12 such that (Oxe , E)∩(G, ∗ ∗ ˜ ˜ Hence, xe 6∈(G, E)12 . Similarly, if xe 6∈(G, E)2 , then there exists (Oxe , E) ∈ η2 ⊆ η12 such that ˜ (Oxe , E)∩(G, E) ∈ I˜. Hence, xe 6∈˜ (G, E)∗12 . So, ˜ ˜ (G, E)∗12 ⊆(G, E)∗1 ∩(G, E)∗2 . On the other hand, if xe 6∈˜ (G, E)∗12 , then there exists ˜ E) ∈ I˜. Since (Oxe , E) ∈ η12 (xe ) such that (Oxe , E)∩(G, then there exist (Oxe , E) ∈ η12 (xe ), such that (M1 , E) ∈ η1 , (M2 , E) ∈ η2 ˜ 1 , E) or ˜ 2 , E) implies xe ∈(M (Oxe , E) = (M1 , E)∪(M

83

˜ 2 , E). Now, since (Oxe , E)∩(G, ˜ E) ∈ I˜, then xe ∈(M ˜ ˜ 2 , E)]∩(G, ˜ [(M1 , E)∪(M E) ∈ I which implies that ˜ ˜ ˜ [(M1 , E)∩(G, E)] ∪[(M E)] ∈ I˜. It follows by 2 , E)∩(G, ˜ Definition 2 (1) that (M1 , E)∩(G, E) ∈ I˜ and ˜ ˜ 1 , E), then xe 6∈˜ (G, E)∗1 . ˜ (M2 , E)∩(G, E) ∈ I . So, if xe ∈(M ˜ ˜ 2 , E), then xe 6∈(G, E)∗2 . Hence, xe 6∈˜ (G, E)∗1 If xe ∈(M ˜ ˜ ˜ ∩(G, E)∗2 . Thus, (G, E)∗1 ∩(G, E)∗2 ⊆(G, E)∗12 . ∗ ∗ ˜ Consequently, (G, E)12 = (G, E)1 ∩( G, E)∗2 , ∀ (G, E) ∈ SS(X)E . Some properties of the pairwise soft local function ()∗12 are obtained from the following theorem. Theorem 3.3. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, the pairwise soft local function ()∗12 satisfies the following properties. (1)(φ˜ , E)∗12 = (φ˜ , E). ˜ ˜ (2)(G, E)⊆(H, E) ⇒ (G, E)∗12 ⊆(H, E)∗12 . ∗ ∗ ˜ ˜ ˜ (3)(G, E)12 ∪(H, E)12 ⊆[(G, E)∪(H, E)]∗12 . ˜ (4)[(G, E)∗12 ]∗12 ⊆(G, E)∗12 . (5)(G, E)∗12 is a p-closed soft set, i.e., sclη12 [(G, E)∗12 ] = (G, E)∗12 . ˜ η12 (G, E). (6)sclη12 [(G, E)∗12 ]⊆scl Proof. (1) : From Theorem 2 (1) we have = (φ˜ , E)∗2 = (φ˜ , E). Therefore, (φ˜ , E)∗1 ∗ ∗ ∗ ˜ ˜ ˜ ˜ ˜ φ , E)2 = (φ , E). (φ , E)12 = (φ , E)1 ∩( ˜ ˜ (2) : Let (G, E)⊆(H, E). Then, (G, E)∗1 ⊆(H, E)∗1 and ˜ E)∗2 [ by Theorem 2 (2)]. It follows that (G, E)∗2 ⊆(H, ˜ ˜ ˜ E)∗2 . Hence, E)∗1 ⊆(G, E)∗2 ∩(H, (G, E)∗1 ∩(H, ∗ ∗ ˜ (G, E)12 ⊆(H, E)12 . (3) : It is immediate from (2). (4) : Let (G, E) ∈ SS(X)E . Then, ˜ E)∗12 ]∗2 [(G, E)∗12 ]∗12 = [(G, E)∗12 ]∗1 ∩[(G, ∗ ∗ ∗ ∗ ˜ ˜ ˜ = [(G, E)1 ∩(G, E)2 ]1 ∩[(G, E)1 ∩(G, E)∗2 ]∗2 ˜ ˜ ˜ ˜ ⊆[(G, E)∗1 ]∗1 ∩[(G, E)∗2 ]∗1 ∩[(G, E)∗1 ]∗2 ∩[(G, E)∗2 ]∗2 ∗ ∗ ∗ ∗ ∗ ∗ ˜ ˜ ˜ ˜ ⊆(G, E)1 ∩[(G, E)2 ]1 ∩[(G, E)1 ]2 ∩(G, E)2 ˜ ˜ ⊆(G, E)∗1 ∩(G, E)∗2 = (G, E)∗12 . ˜ Hence, [(G, E)∗12 ]∗12 ⊆(G, E)∗12 . (5) : Let (G, E) ∈ SS(X)E . Then, ˜ η2 (G, E)∗12 sclη12 (G, E)∗12 = sclη1 (G, E)∗12 ∩scl ∗˜ ˜ η2 [(G, E)∗1 ∩(G, ˜ = sclη1 [(G, E)1 ∩(G, E)∗2 ]∩scl E)∗2 ] ˜ η1 (G, E)∗1 ∩scl ˜ η1 (G, E)∗2 ∩scl ˜ η2 (G, E)∗1 ∩scl ˜ η2 (G, E)∗2 ⊆scl ˜ η1 (G, E)∗2 ∩scl ˜ η2 (G, E)∗1 ∩(G, ˜ = (G, E)∗1 ∩scl E)∗2 [byT heorem2(4)] ∗˜ ∗ ∗ = (G, E)1 ∩(G, E)2 = (G, E)12 .

˜ E)∗12 . But, Hence, sclη12 (G, E)∗12 ⊆(G, ∗ ∗ ˜ (G, E)12 ⊆sclη12 (G, E)12 , then sclη12 (G, E)∗12 = (G, E)∗12 . Hence, (G, E)∗12 is a p-closed soft set.

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(6) : By Theorem 2 (4) we have ˜ ηi (G, E), i = 1, 2. (G, E)∗i ⊆scl ˜ η1 (G, E)∩scl ˜ η2 (G, E). It So, (G, E)∗1 ∩˜ (G, E)∗2 ⊆scl ˜ η12 (G, E) [by Corollary 2]. follows that (G, E)∗12 ⊆scl Theorem 3.4. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. The soft mapping sclη∗12 : SS(X)E → SS(X)E , given by ˜ E)∗12 , ∀(G, E) ∈ SS(X)E , is a sclη∗12 (G, E) = (G, E)∪(G, supra soft closure operator and induces the supra soft ∗ given by topology η12 ∗ η12 = {(G, E) ∈ SS(X)E : sclη∗12 (G, E)c = (G, E)c }. Proof. (i) : It is clear by Theorem 3 (1) that and obvious that sclη∗12 (φ˜ , E) = (φ˜ , E) ˜ η∗ (G, E). (G, E)⊆scl 12 ˜ ˜ (ii) : If (G, E)⊆(H, E), then (G, E)∗12 ⊆(H, E)∗12 ∗ ∗ ˜ ˜ ˜ implies (G, E)∪(G, E)12 ⊆(H, E) ∪(H, E)12 . Hence, ˜ η∗ (H, E). sclη∗12 (G, E)⊆scl 12 (iii) : By (ii), for any two soft sets (G, E), (H, E) we ˜ η∗12 deduce that sclη∗12 (G, E)∪scl ∗ ˜ η [(G, E)∪(H, ˜ (H, E)⊆scl E)]. 12 Hence, sclη∗12 is a supra soft closure operator. ∗ ˜ E) Consequently, η12 is a supra soft topology on (X, but it is not a soft topology in general. Proposition 3.2. Let (X, η1 , η2 , E, I˜) be a soft ideal ∗ . bitopological space. Then, η12 ⊆ η12 Proof. Enough to prove that ˜ η12 (G, E), ∀ (G, E) ∈ SS(X)E . Let sclη∗12 (G, E)⊆scl ˜ η∗12 (G, E). Then xe ∈(G, ˜ ˜ xe ∈scl E)∪(G, E)∗12 . Since ∗ ˜ η12 (G, E) ˜ η12 (G, E)[by and (G, E)12 ⊆scl (G, E)⊆scl ˜ η12 (G, E). Hence, sclη∗12 (G, Theorem 3 (2)], then xe ∈scl ∗ ˜ η12 (G, E). Consequently, η12 ⊆ η12 E)⊆scl . Example 3.2. Let X = {x, y}, E = {e1 , e2 } and let ˜ E), (G, E)}, η1 = {(φ˜ , E), (X,

Moreover,

η12 ˜ E), (G, E), (H, E), (M, E), (N, E), (P, E)}, {(φ˜ , E), (X, where (P, E) = {(e1 , {x}), (e2 , X)}. Now,

,

˜ E), (G, E), (H, E), (M, E), (N, E), (P, E), (Ki , E), i = 1, 2, 3, 4, 5, 6, 7, 8, 9} SS(X)E = {(φ˜ , E), (X,

where (K1 , E) = {(e1 , φ ), (e2 , {x})}, (K2 , E) = {(e1 , φ ), (e2 , X)}, (K3 , E) = {(e1 , {x}), (e2 , φ )}, (K4 , E) = {(e1 , {y}), (e2 , φ )}, (K5 , E) = {(e1 , {y}), (e2 , {y})}, (K6 , E) = {(e1 , {y}), (e2 , {x})}, (K7 , E) = {(e1 , X), (e2 , φ )}, (K8 , E) = {(e1 , X), (e2 , {x})}, (K9 , E) = {(e1 , X), (e2 , {y})}. Let I˜ = {(φ˜ , E), (I1 , E), (I2 , E), (I3 , E)}, where (I1 , E) = {(e1 , φ ), (e2 , {y})}, (I2 , E) = {(e1 , {y}), (e2 , φ )}, (I2 , E) = {(e1 , {y}), (e2 , {y})}. Then, (X, η1 , η2 , E, I˜) is a soft ideal bitopological space. Now, ξ (X)E = {xe1 , xe2 , ye1 , ye2 }, and

(G, E) = {(e1 , {x}), (e2 , {x})},

η12 (xe1 ) = {(G, E), (M, E), (P, E), (X˜ , E)}, η12 (xe2 ) = {(G, E), (N, E), (P, E), (X˜ , E)}, ˜ E)}, η12 (ye1 ) = {(N, E), (X, η12 (ye2 ) = {(H, E), (M, E), (N, E), (P, E), (X˜ , E)}.

and

∗ . ˜ E) ∈ η12 Clear that (φ˜ , E), (X,

where

˜ E), (H, E), (M, E), (N, E)}, η2 = {(φ˜ , E), (X, where (H, E) = {(e1 , φ ), (e2 , {y})}, (M, E) = {(e1 , {x}), (e2 , {y})}, (N, E) = {(e1 , {y}), (e2 , X)}. Then, (X, η1 , η2 , E) is a soft bitopological space.

c 2018 NSP


=

Now, from definition of ()∗12 -operator and the fact ˜ E)∗12 we get the following = (F, E)∪(F, results: sclη∗12 (F, E)

∗

(G, E)c12 = (φ , E) and so sclη∗12 (G, E)c = (G, E)c . ∗ , Hence, (G, E) ∈ η12 ∗

(H, E)c12 = (H, E)c and so sclη∗12 (H, E)c = (H, E)c . ∗ , Hence, (H, E) ∈ η12


∗

∗ = {(G, E) = (G , E)∪(G ˜ 2 , E) : (Gi , E) ∈ ηi∗ , i = 1, 2} η12 1

(M, E)c12 = (M, E)c and so sclη∗12 (M, E)c = (M, E)c . ∗ , Hence, (M, E) ∈ η12

Then, Proof. Let (G, E) ∈ η ∗ 12 . sclη∗12 (G, E)c = (G, E)c implies ˜ η∗2 (G, E)c [ by Theorem 3]. (G, E)c = sclη∗1 (G, E)c ∩scl c ∗ c ˜ Therefore, (G, E) = [sintη∗1 (G, E)]c ∩[sint η2 (G, E)] ∗ c c ∗ [for sintηi (G, E) = [sclηi (G, E) ] , i = 1, 2.]. Hence, ∗ ˜ Take (G, E) = sintη∗1 (G, E)∪sint η2 (G, E). ∗ (G1 , E) = sintη1 (G, E), (G2 , E) = sintη∗2 (G, E).

∗

(N, E)c12 = (N, E)c and so sclη∗12 (N, E)c = (N, E)c . ∗ , Hence, (N, E) ∈ η12 ∗

(P, E)c12 = (φ , E) and so sclη∗12 (P, E)c = (P, E)c . ∗ , Hence, (P, E) ∈ η12 ∗

(K1 , E)c12 = {(e1 , {x}), (e2 , φ )} and ∗ , ∗ sclη12 (K1 , E)c = (K1 , E)c . Hence, (K1 , E) ∈ η12 ∗ (K2 , E)c12 ∗ sclη12 (K2 , E)c

so

= {(e1 , {x}), (e2 , φ )} and ∗ , = (K2 , E)c . Hence, (K2 , E) ∈ η12

so

(K3 , E)c12 = {(e1 , {y}), (e2 , {x})} and ∗ , sclη∗12 (K3 , E)c = (K3 , E)c . Hence, (K3 , E) ∈ η12

so

∗

∗ (K4 , E)c12 sclη∗12 (K4 , E)c ∗

∗

(K6 , E)c12 = {(e1 , {x}), (e2 , φ )} and ∗ , ∗ sclη12 (K6 , E)c = (K6 , E)c . Hence, (K6 , E) ∈ η12 ∗ (K7 , E)c12 ∗ sclη12 (K7 , E)c

˜ 2 , E) such that Conversely, let (G, E) = (G1 , E)∪(G (Gi , E) ∈ ηi∗ , i = 1, 2. Then, ˜ 2 , E)c ] sclη∗12 (G, E)c = sclη∗12 [(G1 , E)c ∩(G ∗ c˜ ˜ η∗2 [(G1 , E)c ∩(G ˜ 2 , E)c ] = sclη1 [(G1 , E) ∩(G2 , E)c ]∩scl ˜ η∗ (G1 , E)c ∩scl ˜ η∗1 (G2 , E)c ∩scl ˜ η∗2 (G1 , E)c ∩scl ˜ η∗2 (G2 , E)c ⊆scl 1 ∗ c˜ c˜ ∗ c˜ = (G1 , E) ∩sclη1 (G2 , E) ∩sclη2 (G1 , E) ∩(G2 , E)c ˜ 2 , E)c = (G, E)c . = (G1 , E)c ∩(G

= {(e1 , X), (e2 , {x})} and so ∗ , ˜ E) 6= (K4 , E)c . Hence, (K4 , E) 6∈ η12 = (X,

(K5 , E)c12 = {(e1 , X), (e2 , {x})} and ∗ , sclη∗12 (K5 , E)c 6= (K5 , E)c . Hence, (K5 , E) 6∈ η12

= {(e1 , {y}), (e2 , {x})} and ∗ 6= (K7 , E)c . Hence, (K7 , E) 6∈ η12 ,

˜ Hence, sclη∗12 (G1 , E)c ⊆(G, E)c . But, c c ∗ ˜ Therefore, (G, E) ⊆sclη12 (G1 , E) . ∗ . This sclη∗12 (G, E)c = (G, E)c . Consequently, (G, E) ∈ η12 complete the proof. Remark 3.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, ˜ ˜ (1)[(G, E)∪(H, E)]∗12 6= (G, E)∗12 ∪(H, E)∗12 in general. ∗ ∗ ˜ ˜ η∗12 (H, E) in (2)sclη12 [(G, E)∪(H, E)] 6= sclη12 (G, E)∪scl general. ∗ which induced by scl ∗ (3)η12 η12 is not soft topology in general.

so so so

∗ (K8 , E)c12 = (φ˜ , E) and so sclη∗12 (K8 , E)c = (K8 , E)c . ∗ , Hence, (K8 , E) ∈ η12 ∗ (K9 , E)c12 ∗ sclη12 (K9 , E)c

= {(e1 , {y}), (e2 , {x})} and ∗ 6= (K9 , E)c . Hence, (K9 , E) 6∈ η12 .

85

Example 3.3. Let X = {x, y, z, w}, E = {e1 , e2 } and let

so

Consequently, .

∗ ˜ E), (G, E), (H, E), (M, E), (M, E), (N, E), (P, E), (K1 , E), (K2 , E), (K3 , E), (K6 , E), (K8 , E)} η12 = {(φ˜ , E), (X,

∗ . Obvious that η12 ⊆ η12

Theorem 3.4. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, ˜ η∗2 (G, E). sclη∗12 (G, E) = sclη∗1 (G, E)∩scl Proof. Let (G, E) ∈ SS(X)E . Then, ˜ E)∗12 sclη∗12 (G, E) = (G, E)∪(G, ˜ ˜ E)∗2 ] = (G, E)∪[(G, E)∗1 ∩(G, ˜ ˜ ˜ = [(G, E)∪(G, E)∗1 ]∩[(G, E)∪(G, E)∗2 ] ∗ ∗ ˜ η2 (G, E). = sclη1 (G, E)∩scl Theorem 3.5. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space, and let (X, η1 ∗ , η2 ∗ , E) be a soft bitopological space induced by I˜ and the soft local functions ()∗1 , ()∗2 . Then,

I˜ = {(φ˜ , E), (A, E)}, ˜ E), (G, E)}, η1 = {(φ˜ , E), (X, ˜ E), (H, E)}, η2 = {(φ˜ , E), (X, where (A, E) = (e1 , {z}), (e2 , {z}), (G, E) = {(e1 , {x, z}), (e2 , {x, z})}, (H, E) = {(e1 , {y, z, w}), (e2 , {y, z, w})}. Then, (X, η1 , η2 , E, I˜) is a soft ideal bitopological space. ˜ E), (G, E), (H, E)} is a Moreover, η12 = {(φ˜ , E), (X, supra soft topology because ˜ (G, E)∩(H, E) = {(e1 , {z}), (e2 , {z})} 6∈ η12 . Now, let (M, E) = {(e1 , {x}), (e2 , {x})}, (N, E) = {(e1 , {y}), (e2 , {y})}. ˜ E) and Since ye1 , ye2 , ze1 , ze2 , we1 , we2 ∈(H, ˜ (H, E)∩(M, E) = (φ˜ , E) ∈ I˜, then ye1 , ye2 , ze1 , ze2 , we1 , we2 6∈˜ (M, E)∗12 . ˜ E), (G, E)}, It is clear that η12 (xe1 ) = η12 (xe2 ) = {(X, ˜ (G, E)∩(M, E) = (M, E) 6∈ I˜ and ˜ E)∩(M, ˜ ˜ E)∗12 . (X, E) = (M, E) 6∈ I˜. Hence, xe1 , xe2 ∈(M, Consequently, (M, E)∗12 = (M, E).

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By similar way we can easily verify that (N, E)∗12 = {(e1 , {y, w}), (e2 , {y, w})}. We deduce that ˜ (M, E)∗12 ∪(N, E)∗12 = {(e1 , {x, y, w}), (e2 , {x, y, w})}. On the other hand, we can show that ˜ E). ˜ [(M, E)∪(N, E)]∗12 = (X, Hence, ∗ ∗ ˜ ˜ [(M, E)∪(N, E)]12 6= (M, E)12 ∪(N, E)∗12 . Similarly, we can verify ˜ ˜ η∗12 (N, E). E)] 6= sclη∗12 (M, E)∪scl sclη∗12 [(M, E)∪(N,

that

Definition 3.3. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, (G, E) is called a p∗ -open soft ˜ 2 , E), (Gi , E) ∈ ηi∗ , (i = 1, 2). set if (G, E) = (G1 , E)∪(G ∗ The complement of a p -open soft set is a p∗ -closed soft set. Proposition 3.3. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, the family of all p∗ -open sets is a supra soft topology. Moreover, ∗ = {(G, E) ∈ SS(X)E : (G, E) is p∗ -open soft set }. η12 Proof. Straightforward. Theorem 3.6. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. If (M, E) 6∈ I˜, then (M, E) is a p∗ -closed soft set. Proof. Straightforward.

4 p∗ -Separated soft sets Definition 4.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space and let (G, E), (H, E) be two non-null soft sets over X. The soft sets (G, E) and (H, E) are said to be p∗ -separated soft sets if ∗ ˜ = (φ˜ , E), sclη12 (G, E)∩(H, E) ˜ η∗12 (H, E) = (φ˜ , E). (G, E)∩scl 3, we have Example 4.1. In Example (G, E)∗12 = {(e1 , X), (e2 , {x})} and (H, E)∗12 = (φ˜ , E). So, = {(e1 , X), (e2 , {x})} and sclη∗12 (G, E) sclη∗12 (H, E) = (H, E). Furthermore, ˜ E) = (φ˜ , E) and sclη∗12 (G, E)∩(H, ∗ ˜ sclη12 (H, E)∩(G, E) = (φ˜ , E). Hence, (G, E), (H, E) are p∗ -separated soft sets. Proposition 4.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Then, every p-separated soft set is a p∗ -separated soft set. Proof. Straightforward. Proposition 4.2. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space and let (G, E), (H, E) be two non-null ˜ η∗12 (H, E) = (φ˜ , E), soft sets over X. If sclη∗12 (G, E)∩scl then (G, E) and (H, E) are p∗ -separated soft sets. Proof. Straightforward.

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Proposition 4.3. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. If (G, E),(H, E) are p∗ -separated soft ˜ ˜ sets, (M, E)⊆(G, E) and (N, E)⊆(H, E), then (M, E), (N, E) are also p∗ -separated soft sets. Proof.

Since

˜ (N, E)⊆(H, E),

then

˜ η∗ (G, E)∩(H, ˜ ˜ sclη∗12 (M, E)∩(N, E)⊆scl E) = (φ˜ , E). 12 ∗ ˜ Therefore, sclη12 (M, E)∩(N, E) = (φ˜ , E). By similar we ˜ E) = (φ˜ , E). Hence, can prove that sclη∗12 (N, E)∩(M, ∗ (M, E), (N, E) are p -separated soft sets. Proposition 4.4. Every p∗ -separated soft sets are disjoint. Proof. Straightforward. Remark 4.1. The converse of Proposition 4 may not be true as shown by the following example. Example 4.2. In Example 3, we have and (K2 , E) = {(e1 , φ ), (e2 , X)} (K4 , E) = {(e1 , {y}), (e1 , φ )} are disjoint. On the other hand, (K2 , E)∗12 = {(e1 , {y}), (e2 , {x})} and ∗ (K4 , E)12 = (φ˜ , E). It follows that sclη∗12 (K2 , E) = {(e1 , {y}), (e2 , X)} and = (K4 , E). But, sclη∗12 (K4 , E) ˜ 4 , E) 6= (φ˜ , E). Hence, (K2 , E), (K4 , E) sclη∗12 (K2 , E)∩(K are not p∗ -separated soft sets. Proposition 4.5. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space and let (G, E) and (H, E) be two p∗ -open soft sets. Then, (G, E) and (H, E) are p∗ -separated soft sets if and only if (G, E) and (H, E) are disjoint. Proof. The first direction is follows from Proposition 4. Now, suppose that (G, E) and (H, E) are disjoint ∗c . It follows ˜ p∗ -open soft sets, then (G, E)⊆(H, E)c ∈ η12 ˜ E)c implies that sclη∗12 (G, E)⊆(H, ∗ ˜ ˜ sclη12 (G, E)∩(H, E) = (φ , E). By similar way we can ˜ show that sclη∗12 (H, E)∩(G, E) = (φ˜ , E). Hence, (G, E), (H, E) are p∗ -separated soft sets. Proposition 4.6. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space and let (F, E), (M, E) be two p∗ -closed soft sets. Then, (F, E) and (M, E) are p∗ -separated soft sets if and only if (F, E) and (M, E) are disjoint. Proof. By similar to the proof of Proposition 4. Theorem 4.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space, Y ⊆ X and let ˜ E). Then, if (G, E), (H, E) are ˜ Y˜ , E)⊆( ˜ X, (G, E), (H, E)⊆( p∗ -separated soft sets in (X, η1 , η2 , E, I˜), then their are p∗ -separated soft sets in (Y˜ , η1Y , η2Y , E, I˜). Proof. Let (G, E), (H, E) be p∗ -separated soft sets in ˜ E). ˜ Y˜ , E)⊆( ˜ X, (X, η1 , η2 , E, I˜) and let (G, E), (H, E)⊆( Then,


T ∗c sclη∗12Y (G, E) = ˜ {(F, E) : (F, E) ∈ η12Y : ˜ (G, E)⊆(F, E)} T ˜ {(Y˜ , E)∩(M, ˜ = E) : (M, E) ∈ ∗c ˜ ˜ E)} η12 , (G, E)⊆(Y˜ , E)∩(M, T ∗c : (G, E)⊆(M, ˜ ˜ {(Y˜ , E)∩(M, ˜ ˜ ⊆ E) : (M, E) ∈ η12 E)} T ∗c : (G, E)⊆(M, ˜ ˜ ˜ {(M, E) : (M, E) ∈ η12 E)}] = (Y˜ , E)∩[ ˜ η∗12 (G, E). = (Y˜ , E)∩scl

˜ Y˜ , E)∩scl ˜ η∗12 (G, E). It follows Hence, sclη∗12Y (G, E)⊆( ∗ ˜ (Y˜ , E)∩(H, ˜ ˜ that sclη12Y (G, E)∩(H, E)⊆ E)∩˜ Therefore, sclη∗12 (G, E). ∗ ∗ ˜ ˜ ˜ sclη12Y (G, E)∩(H, E)⊆(H, E)∩sclη12 (G, E) = (φ˜ , E). ˜ Consequently, sclη∗12Y (G, E)∩(H, E) = (φ˜ , E). By similar ∗ ˜ way we can prove that sclη12Y (H, E)∩(H, E) = (φ˜ , E). Hence, (G, E), (H, E) are p∗ -separated soft sets in (Y, η1Y , η2Y , E, I˜).

5 Connectedness in soft ideal bitopological spaces Definition 5.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. The p∗ -separated soft sets (G, E), (H, E) in (X, η1 , η2 , E, I˜) are said to be p∗ -soft ˜ E) if (X, ˜ E) = (G, E)∪(H, ˜ separation of (X, E). In this ˜ case we say that (X, E) has a p∗ -soft separation. Definition 5.2. A soft ideal bitopological space (X, η1 , η2 , E, I˜) is said to be a p∗ -soft disconnected ˜ E) has a p∗ -soft separation. Otherwise, space if (X, (X, η1 , η2 , E, I˜) is called p∗ -soft connected space, i.e., ˜ E) (X, η1 , η2 , E, I˜) is a p∗ -soft connected if (X, ∗ cannot be represented as the union of two p -separated soft sets. Theorem 5.1. Let (X, η1 , η2 , E, I˜) be a soft bitopological space. Then, the following are equivalent: (1)(X, η1 , η2 , E, I˜) is a p∗ -soft connected. ˜ E) cannot represented as a union of two non-null (2)(X, disjoint p∗ -open soft sets. ˜ E) cannot represented as a union of two non-null (3)(X, disjoint p∗ -closed soft sets. ˜ E) has no proper soft subset which is both p∗ -open (4)(X, and p∗ -closed soft set other than (φ˜ , E). Proof. Similar to the proof of Theorem 5. Proposition 5.1. Every p∗ - soft connected space is a p-soft connected space. Proof. Straightforward. Definition 5.3. Let (X, η1 , η2 , E) and (Y, σ1 , σ2 , K) be two soft bitopological spaces and let I˜ be a soft ideal on ˜ E). (X, A soft mapping ˜ f pu : (X, η1 , η2 , E, I ) → (Y, σ1 , σ2 , K) is said to be a p∗ -soft continuous if the inverse image of any p-open soft set in (Y, σ1 , σ2 , K) is a p∗ -open soft set in

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(X, η1 , η2 , E, I˜), (H, K) ∈ σ12 .

i.e.,

−1 (H, K) ∈ η ∗ f pu 12

for

any

Theorem 5.2. Let f : (X, η1 , η2 , E, I˜) → (Y, σ1 , σ2 , K) be a p∗ -soft continuous and surjective soft mapping. If (X, η1 , η2 , E, I˜) is a p∗ -soft connected space, then (Y, σ1 , σ2 , K) is a p-connected space. Proof. It is immediate.

6 p∗ -connected(disconnected) soft sets Definition 6.1. A soft set (G, E) in a soft ideal bitopological space (X, η1 , η2 , E, I˜) is said to be a p∗ -connected soft set if cannot expressed as a union of two p∗ -separated soft sets. Otherwise, (G, E) is called a p∗ -disconnected soft set Proposition 6.1. Every p∗ -connected soft set is a p-connected soft set. Proof. Straightforward. Theorem 6.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. If (M, E) is a p∗ -connected soft set and (G, E), (H, E) are p∗ -separated soft sets with ˜ (G, E)∪(H, ˜ ˜ (M, E)⊆ E), then either (M, E)⊆(G, E) or ˜ (M, E)⊆(H, E). Proof. (M, E) =

˜ ˜ Let (M, E)⊆(G, E)∪(H, E). ˜ ˜ ˜ [(M, E)∩(G, E)]∪[(M, E)∩(H, E)],

Since then ˜ ˜ ˜ η∗12 (G, E) = (φ˜ , E). ˜ ˜ η∗12 [(M, E)∩(G, E)]⊆(H, E)∩scl [(M, E)∩(H, E)]∩scl Therefore, ˜ ˜ ˜ η∗12 [(M, E)∩(G, E)] = (φ˜ , E). [(M, E)∩(H, E)]∩scl By similar way, we have ˜ ˜ ˜ η∗12 [(M, E)∩(H, E)] = (φ˜ , E). [(M, E)∩(G, E)]∩scl ˜ ˜ Assume that (M, E)∩(G, E) and (M, E)∩(H, E) are non-null soft sets. Then, (M, E) is not p∗ -connected soft ˜ set, a contradiction. Hence, either (M, E)∩(G, E) = (φ˜ , E) ˜ ˜ ˜ or (M, E)∩(H, E) = (φ˜ , E). But, (M, E)⊆(G, E)∪(H, E), ˜ ˜ then either (M, E)⊆(G, E) or (M, E)⊆(H, E). Theorem 6.2. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. Let (F, E) be a p∗ -connected soft set ˜ ˜ η∗ (F, E). in (X, η1 , η2 , E, I˜) and let (F, E)⊆(M, E)⊆scl 12 ∗ Then, (M, E) is also p -connected soft set. Proof. Suppose that (M, E) is not a p∗ -connected soft set. Then there exist p∗ -separated soft sets (G, E) and ˜ (H, E) such that (M, E) = (G, E)∪(H, E). It follows that (G, E) and (H, E) are non-null soft sets and ˜ η∗12 (G, E) = (φ˜ , E). sclη∗12 (G, E) ∩ (H, E) = (H, E)∩scl ˜ ˜ ˜ But (F, E)⊆(M, E), then (F, E)⊆(G, E)∪(H, E). It ˜ E) or follows by Theorem 6 that either (F, E)⊆(G, ˜ ˜ (F, E)⊆(H, E). We claim that (F, E)⊆(G, E). Then, ˜ η∗ (G, E) which implies that sclη∗12 (F, E)⊆scl 12 (H, E) ∩ sclη∗12 (F, E) = (φ˜ , E). Since ˜ ˜ η∗ (F, E), then (H, E)⊆(M, E)⊆scl 12

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˜ η∗12 (F, E) = (φ˜ , E). Thus, (H, E) is a (H, E) = (H, E)∩scl null soft set, a contradiction. Hence, (M, E) is a p∗ -connected soft set. Corollary 6.1. Let (X, η1 , η2 , E, I˜) be a soft ideal bitopological space. If (F, E) is a p∗ -connected soft set, then sclη∗12 (F, E) is also p∗ -connected soft set. Theorem 6.3. Let (G, E),(H, E) be two p∗ -connected soft sets in soft ideal bitopological space (X, η1 , η2 , E, I˜). If ˜ ˜ E) is a (G, E)∩(H, E) 6= (φ˜ , E), then (G, E)∪(H, p∗ -connected soft set. Proof. Straightforward. Theorem 6.4. Let f pu be a p∗ -soft continuous mapping from a soft ideal bitopoloical space (X, η1 , η2 , E, I˜) into a soft bitopological space (Y, σ1 , σ2 , K). If (G, E) is a p∗ -connected soft set in (X, η1 , η2 , E, I˜), then f pu (G, E) is a p-connected soft set in (Y, σ1 , σ2 , K). Proof. Straightforward. Corollary 6.2. Let f pu be a p∗ -soft continuous mapping from a p∗ -soft connected space (X, η1 , η2 , E, I˜) onto a soft bitopological space (Y, σ1 , σ2 , K). Then (Y, σ1 , σ2 , K) is a p-soft connected space.

Acknowledgements The authors express their sincere thanks to the reviewers for their careful checking of the details and for helpful comments that improved this paper. The authors are also thankful to the editors-in-chief and managing editors for their important comments which helped to improve the presentation of the paper.

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Ali Kandil Saad Ibrahim is a Professor of Mathematics at Helwan University. He received the Ph.D. degree in Topology from the University of Moscow in 1978. His primary research areas are General Topology, Fuzzy Topology, double sets and theory of sets. Dr. Kandil has published over 120 papers in refereed journals and contributed several book chapters in various types of Mathematics textbooks. He is a Fellow of the Egyptian Mathematical Society and Egyptian Physics Mathematical Society. He was the Supervisor of 25 PHD and about 32 MSC students. Osama Abd El-Hamid El-Tantawy is a Professor of Mathematics at Zagazig University. He was born in 1951. He received the Ph.D. degree in Topology from the University of Zagazig in 1988. His primary research areas are General Topology, Fuzzy Topology, double sets and theory of sets. Dr. Osama has published over 80 papers in refereed journals. He is a Fellow of the Egyptian Mathematical Society and Egyptian Physics Mathematical Society. He was the Supervisor of 15 PHD and about 20 MSC students.

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Sobhy Ahmed Aly El-Sheikh is a Professor of pure Mathematics, Ain Shams University, Faculty of Education, Mathematic Department, Cairo, Egypt. He was born in 1955. He received the Ph.D. degree in Topology from the University of Zagazig. His primary research areas are General Topology, Fuzzy Topology, double sets and theory of sets. Dr. Sobhy has published over 85 papers in Fuzzy set and system Journal (FSS), Information science Journal (INFS), Journal of fuzzy Mathematics, and other refereed journals. He is a Fellow of the Egyptian Mathematical Society. He was the Supervisor of 12 PHD and about 18 MSC students.

Shawqi Ahmed Hazza Al-Alimi is a Lecturer of Pure Mathematics, Department of Mathematics, Faculty of Education, Taiz University, Taiz, Yemen. He received the MSc degree in Pure Mathematics (Topology) from Ain Shams University, Cairo, Egypt. His research interests are in the areas of General Topology and Rough Sets. She has published more than 13 research articles in reputed international journals.

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