EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 199-210 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Some Properties of Weak Soft Axioms Sabir Hussain Department of Mathematics, College of Science, Qassim University,Buraydah 51482, Saudi Arabia Abstract. The aim of this paper is to initiate and discuss the properties and characterizations of soft semi-Ti and soft semi-Di( for i = 0, 1, 2) spaces at soft point by analyzing the relationship among them. We also introduce and explore the properties of soft S-continuous functions. These results will be useful to enhance the theoretical framework and to promote further study towards the daily life applications. 2010 Mathematics Subject Classifications: 06D72, 54A10, 54D10 Key Words and Phrases: Soft sets, soft topology, soft semi-open(closed), soft semi-closure, soft semi-Di (for i = 0, 1, 2) spaces, soft semi-Ti(for i = 0, 1, 2 ) spaces, soft S-continuous function 1. Introduction An application of soft sets in decision making problems that is based on the reduction of parameters to keep the optimal choice objects can be seen. This is also useful in the process to construct models during the modelling process in different fields of life. The real world is inherently uncertain, imprecise and vague. Because of various uncertainties, classical methods are not successful for solving complicated problems in economics, engineering and environment. A soft set is a collection of approximate descriptions of an object and is free from the parameterizations inadequacy syndrome of fuzzy set theory, rough set theory, probability theory and game theory. Soft systems provide a very general framework with the involvement of parameters. Research works on soft set theory, its generalized structures and its applications in various fields are progressing rapidly now a days. Molodtsov [14, 15] initiated and applied soft sets theory, while modelling the problems in the field of science including engineering physics, computer science, economics, social sciences and medical sciences, to deal with uncertain data and not clear objects without complete information. Maji et al. [12, 13] discussed and applied the soft set theory in decision making problems. In [17] and [19], Xiao et al. and Pei et al. respectively explored the soft sets in information systems. The criteria of measuring the sound quality through the soft sets studied by Kostek [11]. Mushrif et al. [16] established the remarkable method for the classification of natural textures by applying the concept of soft set. Email addresses: sabiriub@yahoo.com; sh.hussain@qu.edu.sa (S. Hussain) http://www.ejpam.com 199 c© 2017 EJPAM All rights reserved. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 200 In [18], Shabir and Naz introduced and studied the primary concepts of soft topological spaces. After that Hussain [6, 7], Hussain and Ahmad [8, 9] and [1], Aygunoglu et al. [2], Zorlutana et al. [20] continued to add many basic concepts in soft topological spaces. In [3, 4], Chen introduced and explored soft semi-open(closed) sets in soft topological spaces. In [5], Hussain added many concepts toward soft semi-open sets and soft semi-closed sets in soft topological spaces. Kharral and Ahmad [10] and then Zorlutana [20] discussed the mappings of soft classes and their properties in soft topological spaces. Recently in [7], Hussain presented and discussed basic properties and characterizations of soft pu-continuous functions and soft pu-open(closed) functions. 2. Preliminaries First we recall some definitions and results which will use in the sequel. Definition 1 ([14]). Let X be an initial universe and E be a set of parameters. Let P (X) denotes the power set of X and A be a non-empty subset of E. A pair (F,A) is called a soft set over X, where F is a mapping given by F : A→ P (X). In other words, a soft set over X is a parameterized family of subsets of the universe X. For e ∈ A, F (e) may be considered as the set of e-approximate elements of the soft set (F,A). Clearly, a soft set is not a set. Here we consider only soft sets (F,A) over a universe X in which all the parameters of set A are same. We denote the family of these soft sets by SS(X)A. For soft subsets, soft union, soft intersection, soft complement, their properties and the relations to each other; the interested reader is refer to [12, 13, 14, 15]. Definition 2 ([18]). Let τ be the collection of soft sets over X, then τ is said to be a soft topology on X, if (1) Φ, X̃ belong to τ . (2) the union of any number of soft sets in τ belongs to τ . (3) the intersection of any two soft sets in τ belongs to τ . The triplet (X, τ,E) is called a soft topological space over X. Every member of τ is called soft open set. A soft set is called soft closed if and only if its complement is soft open. Definition 3 ([8, 18]). Let (X, τ,E) be a soft topological space over X and A ⊆ X. Then (1) soft interior of soft set (F,A) over X denoted by (F,A)◦ and is defined as the union of all soft open sets contained in (F,A). Thus (F,A)◦ is the largest soft open set contained in (F,A). (2) soft closure of (F,A), denoted by (F,A) is the intersection of all soft closed super sets of (F,A). Clearly (F,A) is the smallest soft closed set over X which contains (F,A). S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 201 (3) soft boundary of soft set (F,A) over X denoted by (F,A) and is defined as (F,A) = (F,A) ∩ ((F,A)′). Obviously (F,A) is a smallest soft closed set over X containing (F,A). For detailed properties of soft interior, soft closure and soft boundary, we refer to [8]. Definition 4 ([3]). Let (X, τ,E) be a soft topological space over X with A ⊆ X and (F,A) be a soft set over X. Then (F,A) is called soft semi-open set if and only if there exists a soft open set (G,A) such that (G,A)⊆̃(F,A)⊆̃(G,A). The set of all soft semi-open sets is denoted by S.S.O(X). Note that every soft open set is soft semi-open set. A soft set (F,A) is said to be soft semi-closed if its soft relative complement is soft semi-open. Equivalently, there exists a soft closed set (G,A) such that (G,A)◦⊆̃(F,A)⊆̃(G,A). Note that every soft closed set is soft semi-closed set. Definition 5 ([5]). Let (X, τ,E) be a soft topological space over X with A ⊆ X. [(i)]soft semi-interior of soft set (F,A) over X denoted by ints(F,A) and is defined as the union of all soft semi-open sets contained in (F,A). soft semi-closure of (F,A) over X denoted by cls(F,A) is the intersection of all soft semi-closed super sets of (F,A). For detailed properties of soft semi-open(closed) and soft semi-interior(closure) we refer to [3, 4, 5]. 3. Soft Semi-Separation Axioms Hereafter, SS(X)A denotes the family of soft sets over X with the set of parameters A. Definition 6 ([13]). A soft set (F,A) over X is said to be an absolute soft set, denoted by X̃A, if for all e ∈ A, F (e) = X. Clearly, X̃c A = ΦA and Φc A = X̃A. Definition 7 ([13]). A soft set (F,A) over X is said to be null soft set, denoted by Φ̃A, if for all e ∈ A, F (e) = φ. Proposition 1 ([20]). Let eF ∈̃X̃A and (G,A)∈̃SS(X)A. If eF ∈̃(G,A), then eF /̃∈(G,A)c. Definition 8 ([20]). The soft set (F,A)∈̃SS(X)A is called soft point in X̃A, denoted by eF , if for the element e ∈ A, F (e) 6= φ and F (e ′ ) = φ, for all e ′ ∈ A− {e}. Definition 9 ([20]). The soft point eF is said to be in the soft set (G,A), denoted by eF ∈̃(G,A), if for the element e ∈ A, F (e) ⊆ G(e). Definition 10 ([6]). Two soft sets (G,A), (H,A) in SS(X)A are said to be soft disjoint, written (G,A)∩̃(H,A) = ΦA, if G(e) ∩H(e) = φ, for all e ∈ A. Definition 11 ([6]). Two soft points eG, eH in X̃A are distinct, written eG 6= eH , if there corresponding soft sets (G,A) and (H,A) are soft disjoint. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 202 Definition 12. Let (X, τ,A) be a soft topological space over X and (F,A)∈̃SS(X)A. Then (F,A) is soft semi-D-set, if there exists two soft semi-open sets (G,A) and (H,A) such that (G,A) ˜6=X̃ and (F,A)=̃(G,A)\̃(H,A). From the definition, it is clear that every soft semi-open set (G,A) ˜6=X̃ is soft semi-D- set, if (F,A)=̃(G,A) and (H,A)=̃Φ̃. Example 1. Let X = {h1, h2, h3}, A = {e1, e2} and τ = {Φ̃, X̃, (K1, A), (K2, A), (K3, A), (K4, A), (K5, A), (K6, A), (K7, A)} where (K1, A), (K2, A), (K3, A), (K4, A), (K5, A), (K6, A) and (K7, A) are soft sets over X, defined as follows: K1(e1) ={h1, h2},K1(e2) = {h1, h2},K2(e1) = {h2},K2(e2) = {h1, h3},K3(e1) = {h2, h3}, K3(e2) ={h1},K4(e1) = {h2},K4(e2) = {h1},K5(e1) = {h1, h2},K5(e2) = X,K6(e1) = X, K6(e2) ={h1, h2},K7(e1) = {h2, h3},K7(e2) = {h1, h3}. Then τ defines a soft topology on X and hence (X, τ,A) is a soft topological space over X. Clearly (F,A)=̃{{h1}, {h2, h3}} is a soft semi-D-set, because (F,A)=̃(K5, A)\̃(K4, A). Similarly, (K,A)=̃{{h1}, {h2}} is soft semi-D-set, because (K,A)=̃(K1, A)\̃(K2, A). Definition 13. Let (X, τ,A) be a soft topological space over X. Then (X, τ,A) is called soft semi-D0 space, if for any two distinct soft points eF and eG in X̃A, there exists soft semi-D-set (H,A) in SS(X)A such that eF ∈̃(H,A) and eG /̃∈(H,A) or soft semi-D-set (K,A) in SS(X)A such that eG∈̃(K,A) and eF /̃∈(K,A). Definition 14. Let (X, τ,A) be a soft topological space over X. Then (X, τ,A) is called soft semi-D1 space, if for any two distinct soft points eF and eG in X̃A, there exists soft semi-D-set (H,A) in SS(X)A such that eF ∈̃(H,A) and eG /̃∈(H,A) and soft semi-D-set (K,A) in SS(X)A such that eG∈̃(K,A) and eF /̃∈(K,A). Definition 15. Let (X, τ,A) be a soft topological space over X. Then (X, τ,A) is called soft semi-D2 space, if for any two distinct soft points eF and eG in X̃A, there exists disjoint soft semi-D-sets (H,A) and (K,A) in SS(X)A such that eF ∈̃(H,A) and eG∈̃(K,A). Remark 1. It is clear from the above definitions that soft semi-D2 ⇒ soft semi-D1 ⇒ soft semi-D0. The interested reader can easily check that the converse is not true in general. Definition 16. Let (X, τ,A) be a soft topological space over X. If for any two distinct soft points eG, eH in X̃A, there exist soft semi-open sets (F1, A) or (F2, A) such that eG∈̃(F1, A), eH /̃∈(F1, A), eH ∈̃(F2, A), eG /̃∈(F2, A), then (X, τ,A) is called soft semi-T0- space. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 203 Example 2. Let X = {h1, h2, h3}, A = {e1, e2, e3} and τ = {Φ̃, X̃, (K1, A)}, where (K1, A)=̃{(e1, {h1}), (e2, {h2}), (e3, {h3})} is a soft sets over X with soft points: e1(K1) = {h1}, e2(K1) = {h2}, e3(K1) = {h3}. Then τ defines a soft topology on X and hence (X, τ,A) is a soft topological space over X. Moreover (X, τ,A) is soft semi-T0-space. Definition 17. Let (X, τ,A) be a soft topological space over X. (G,A) be soft set in SS(X)A, and eF be a soft point in X̃A. Then (G,A) is said to be soft semi-neighborhood of soft point eF , if there exists a soft open set (K,A) such that eF ∈̃(K,A)⊂̃(G,A). Definition 18. Let (X, τ,A) be a soft topological space over X. (F,A) be soft set in SS(X)A, and eF be a soft point in X̃A. If every soft neighborhood of eF soft intersects (F,A) in some soft points other than eF itself, then eF is called soft semi-limit point of (F,E). The set of all soft semi-limit points of (F,A) is denoted by (F,A)ssd. In other words, if (X, τ,A) is a soft topological space, (F,A) be soft set in SS(X)A, and eF be soft point in X̃A, then eF ∈̃(F,A)ssd if and only if (G,A)∩̃((F,A)\̃{eF }) ˜6=Φ̃, for all soft semi-open neighborhoods (G,A) of eF . Remark 2. Form the definition, it follows that the soft point eF is a soft semi-limit point of (F,A) if and only if eF ∈̃cls((F,A)\̃{eF }). Theorem 1. Let (X, τ,A) be a soft topological space over X. Then the following are equivalent: (1) (X, τ,A) is soft semi-T0 space. (2) For any distinct soft points eG and eH in X̃A, cls(eG) ˜6=cls(eH). Proof. (1) ⇒ (2) Suppose that (X, τ,A) is soft semi-T0 space and eG and eH are distinct soft points in X̃A. Then there exists at least one soft semi-open set (K,A) (say), which contains eG but not eH . But then eG is not soft semi-limit point of eH . Since eG is not in eH , cls(eG) ˜6=cls(eH). (2)⇒ (1) Suppose that for any distinct soft points eG and eH in XA, cls(eG) ˜6=cls(eH). Contrarily suppose that X̃A is not soft semi-T0 space. Then every soft semi-open set which contains eG also contains eH . Then by the property of soft semi-limit point, eG is in cls(eH) so that cls(eG)⊆̃cls(eH). Similarly every soft semi-pen set which contains eH also contains eG (otherwise X̃A would be a soft semi-T0 space). So cls(eH)⊆̃cls(eG). Thus cls(eG)=̃cls(eH). This contradiction proves as required. Definition 19. Let (X, τ,A) be a soft topological space over X and eG, eH are two dis- tinct soft points in X̃A. If there exists a soft semi-open set (F1, A) such that eG∈̃(F1, A), eH /̃∈(F1, A) and a soft semi-open set (F2, A) such that eH ∈̃(F2, A), eG /̃∈(F2, A), then (X, τ,A) is called soft semi-T1-space. Example 3. In Example 2, (X, τ,A) is not soft semi-T1 space. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 204 Theorem 2. Let (X, τ,A) be a soft topological space over X. Then the following are equivalent to each other: (1) (X, τ,A) is soft semi-T1. (2) {eF } is soft semi-closed, for each soft point eF in X̃A. Proof. (1) ⇒ (2) Let (X, τ,A) be a soft topological space over X. Let eF be soft point in X̃A and eG ∈ {eF }c. Then eF and eG are distinct soft points. Since (X, τ,A) is a soft semi-T1 space, then there exists a soft semi-open set (H,A) with eG∈̃(H,A) and eF /̃∈(H,A). Thus, eG∈̃(H,A)⊆̃{eF }c. This follows that {eF }c can be written as the soft union of soft semi-open sets (Hi, A) with eHi∈̃{eF }c. Hence {eF }c is soft semi-open which implies that {eF } is soft semi-closed. (2)⇒ (1) Let {eF } is soft semi-closed, for each soft point eF in X̃A. Suppose eF and eG be distinct soft points in X̃A. Then eG∈̃{eF }c. Thus {eF }c is a soft semi-open set with eG∈̃{eF }c and eF /̃∈{eF }c. Also {eG}c is a soft semi-open set with eF ∈̃{eG}c and eG /̃∈{eG}c. This implies that (X, τ,A) is soft semi-T1 space. Definition 20. Let (X, τ,A) be a soft topological space over X and eG, eH are two distinct soft points in X̃A. If there exist soft disjoint soft semi-open sets (F1, A) and (F2, A) such that eG∈̃(F1, A), eH ∈̃(F2, A), then (X, τ,A) is called soft semi-T2-space. Example 4. Let (X, τ,A) be a soft discrete soft topological space [18]. Then (X, τ,A) is soft semi-T2 space. Theorem 3. Let (X, τ,A) be a soft topological space over X and eG, eH are distinct soft points in X̃A. Then (X, τ,A) is soft semi-T2-space, implies that there exist soft semi-closed sets (H,A) and (K,A) such that eG∈̃(H,A), eH /̃∈(H,A) and eG /̃∈(K,A), eH ∈̃(K,A), and (H,A)∪̃(K,A) = X̃A. Proof. Since (X, τ,A) is soft semi-T2-space and eG and eH are distinct soft points in X̃A, then there exist soft disjoint soft semi-open sets (G1, A) and (G2, A) such that eG∈̃(G1, A) and eH ∈̃(G2, A). Clearly (G1, A)⊆̃(G2, A)c and (G2, A)⊆̃(G1, A)c. Hence eG∈̃(G2, A)c. Put (G2, A)c = (H,A). This gives eG∈̃(H,A) and eH /̃∈(H,A). Also eH ∈̃(G1, A)c. Put (G1, A)c=̃(K,A). Therefore eG∈̃(H,A) and eH ∈̃(K,A). Moreover (H,A)∪̃(K,A)=̃(G2, A)c∪̃(G1, A)c=̃X̃A. Remark 3. From the above theorem and by definitions of soft semi-Di and soft semi- Ti(for i = 0, 1, 2) spaces, clearly we have: (1) soft semi-T2 ⇒ soft semi-T1 ⇒ soft semi-T0 (2) soft semi-Ti ⇒ soft semi-Di (for i = 0, 1, 2). S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 205 Theorem 4. Let (X, τ,A) be a soft topological space over X and eF , eG are distinct soft points in X̃A. Then (X, τ,A) is soft semi-D0 space if and only if (X, τ,A) is soft semi-T0 space. Proof. (⇒) Let (X, τ,A) be a soft semi-D0 space. Then for each distinct soft point eF , eG in X̃A, there exists a soft semi-D-set (H,A) in SS(X)A such that eF ∈̃(H,A) and eG /̃∈(H,A). Suppose that (H,A)=̃(F,A)\̃(G,A) , where (F,A) and (G,A) are soft semi- open sets and (F,A) ˜6=X̃. This follows that eF ∈̃(F,A) and for eG /̃∈(H,A), we have two possibilities: [1.]eG /̃∈(F,A). Therefore, eF ∈̃(F,A) and eG /̃∈(F,A). eG∈̃(F,A) and eG∈̃(G,A). Hence eG∈̃(G,A) and eF /̃∈(G,A). This follows that (X, τ,A) is soft semi-T0 space. (⇐) The proof follows from Remark 3(2). Theorem 5. Let (X, τ,A) be a soft topological space over X and eF , eG are distinct soft points in X̃A. Then (X, τ,A) is soft semi-D1 space if and only if (X, τ,A) is soft semi-D2 space. Proof. (⇒) Let (X, τ,A) be soft semi-D1 space. Then for any two distinct soft points eF and eG in X̃A, there exists soft semi-D-sets (H,A) and (K,A) in SS(X)A such that eF ∈̃(H,A), eG /̃∈(H,A), eG∈̃(K,A), eF /̃∈(K,A). Consider soft sets (F,A), (G,A), (L,A) and (M,A) such that (H,A)=̃(F,A)\̃(G,A) and (K,A)=̃(L,A)\̃(M,A). eF /̃∈(K,A), im- plies that either eF /̃∈(L,A) or eF ∈̃(L,A) and eF ∈̃(M,A). We suppose two cases: [Case (1).]If eF /̃∈(L,A). As eG∈̃(H,A) then either eG∈̃(F,A) and eG∈̃(G,A) or eG /̃∈(F,A). If eG∈̃(F,A) and eG∈̃(G,A). Then eF ∈̃(F,A)\̃(G,A), eG∈̃(G,A) and ((F,A)\̃(G,A))∩̃(G,A)=̃Φ̃. If eG /̃∈(F,A). As eF ∈̃(F,A)\̃(G,A), we have that eF ∈̃(F,A)\̃((G,A)∪̃(L,A)) and from eG∈̃(L,A)\̃(M,A), we have eG∈̃(L,A)\̃((F,A)∪̃(M,A)). Clearly ((F,A)\̃((G,A)∪̃(L,A)))∩̃((L,A)\̃((F,A)∪̃(M,A)))=̃Φ̃. If eF ∈̃(L,A) and eF ∈̃(M,A). Then eG∈̃(L,A)\̃(M,A), eF ∈̃(M,A) and ((L,A)\̃(M,A))∩̃(M,A)=̃Φ̃. Thus in each case, (X, τ,A) is soft semi-D2 space. (⇐) This follows from Remark 1. Hence the proof. Proposition 2. Let (X, τ,A) be a soft topological space over X. If (X, τ,A) is soft semi- D1 space, then X is soft semi-T0 space. Proof. The proof follows directly form Remark 1 and Theorem 4. Definition 21. Let (X, τ,A) be a soft topological space over X and eF , eG be any soft points in X̃A. If eF ∈̃cls({eG}) implies eG∈̃cls({eF }), then (X, τ,A) is called soft semi- symmetric. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 206 Definition 22. Let (X, τ,A) be a soft topological space over X and (F,A) be a soft set in SS(X)A. If for any soft semi-open set (H,A) in SS(X)A and (F,A)⊆̃(H,A) implies cls(F,A)⊆̃(H,A), then (F,A) is called soft semi-generalized closed (in short soft sg-closed) set. The proof of the following proposition is straightforward form the definition of soft semi-closed and soft sg-closed set. Proposition 3. In a soft topological space (X, τ,A) over X, every soft semi closed set (F,A) is soft sg-closed. Theorem 6. Let (X, τ,A) be a soft topological space over X. Then the following state- ments are equivalent: (1) {eF } is soft sg-closed, for any soft point eF in X̃A. (2) (X, τ,A) is soft semi-symmetric. Proof. (1)⇒ (2) Assume that eF ∈̃cls({eG}). Suppose on the contrarily that eG /̃∈cls({eF }). Then eG∈̃(cls({eG}))c. This follows that {eG}⊆̃(cls({eF }))c. Therefore, cls({eG})⊆̃(cls({eF }))c. Hence eG∈̃(cls({eF }))c. This contradiction proves the required result. (2) ⇒ (1) Assume on the contrary that for soft point eF in X̃A and a soft semi- open set (H,A) in SS(X)A such that {eF }⊆̃(H,A) and cls({eF }) ˜6⊆(G,A). This fol- lows that cls({eF })∩̃(H,A)c ˜6=Φ̃. Let us take a soft point eG in X̃A and assume that eG∈̃(cls({eF })∩̃(H,A)c). Here we have eF ∈̃cls({eG}). This implies that cls({eG})⊆̃(H,A)c and eF /̃∈(H,A). A contradiction. Hence the proof. Theorem 7. Any soft semi-T1 space is soft semi-symmetric in a soft topological space (X, τ,A) over X. Proof. Let (X, τ,A) be soft semi-T1 space. Then Theorem 2 follows that {eF } is soft semi-closed, for any soft point eF in X̃A. Thus {eF } is soft sg-closed, by Proposition 3. Therefore Theorem 6 implies that {eF } is soft semi-symmetric. This completes the proof. Theorem 8. Let (X, τ,A) be a soft topological space over X. Then (X, τ,A) is a soft semi-symmetric and soft semi-T0 space if and only if it is soft semi-T1. Proof. (⇒) Suppose (X, τ,A) is soft semi-T0 space. Then for any two distinct soft point eF and eG in X̃A, there exists soft semi-open set (H,A) in SS(X)A such that eF ∈̃(H,A)⊆̃({eG})c. This implies that eF /̃∈cls({eG}). Therefore, eG /∈ cls({eF }). This follows that there exists a soft semi-open set (K,A) such that eG∈̃(K,A)⊆̃({eF })c. There- fore (X, τ,A) is soft semi-T1 space. (⇐) Using Theorem 6 and Remark 3(1), proof follows directly. Hence the proof. The following theorem follows from Remark 3(1), Theorem 5, Proposition 2 and The- orem 8. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 207 Theorem 9. If a soft topological space (X, τ,A) over X is soft semi-symmetric, then we have: (X, τ,A) is soft semi-T0 space ⇔ (X, τ,A) is soft semi-D1 space ⇔ (X, τ,A) is soft semi-T1 space. 4. Properties of Soft S-Continuous Functions Definition 23 ([10]). Let SS(X)A and SS(Y )B be two families of soft sets. u : X → Y and p : A→ B be mappings. Then the image and the inverse image of a function fpu : SS(X)A → SS(Y )B is defined as follows: (1) Let (F,A) be soft set in SS(X)A. The image of (F,A) under fpu, written as fpu(F,A) = (fpu(F ), p(A)), is a soft set in SS(Y )B such that fpu(F )(y) = {⋃ x∈p−1(y)∩A u(F (x)), p−1(y) ∩A 6= φ φ, otherwise for all y ∈ B. (2) Let (G,B) be soft set in SS(Y )B. Then the inverse image of (G,B) under fpu, written as f−1pu (G,B) = (f−1pu (G), p−1(B)), is a soft set in SS(X)A such that f−1pu (G)(x) = { u−1(G(p(x))), p(x) ∈ B φ, otherwise for all x ∈ A. The soft function fpu is called soft surjective, if p and u are surjective. The soft function fpu is called soft injective, if p and u are injective. For detailed properties of soft functions, we refer to [7, 10, 20]. Definition 24. Let (X, τ,A) and (Y, τ∗, B) be soft topological spaces over X and Y respectively and u : X → Y and p : A → B be mappings. Then the soft function fpu : SS(X)A → SS(Y )B is soft S-continuous, if for any soft semi-open set (G,B) in SS(Y )B, f−1pu (G,B)) is soft semi-open in SS(X)A. Theorem 10. Let (X, τ,A) and (Y, τ∗, B) be soft topological spaces over X and Y respec- tively. If a soft function fpu : SS(X)A → SS(Y )B is soft surjective soft S-continuous, then for each soft semi-D set (G,B) in SS(Y )B, f−1pu (G,B) is soft semi-D set in SS(X)A. Proof. Suppose that soft function fpu is soft surjective soft S-continuous and (G,B) be soft semi-D set in SS(Y )B. Then there exist soft semi-open sets (H,B) and (K,B) in SS(Y )B such that (H,B) ˜6=Ỹ and (G,B)=̃(H,B)\̃(K,B). Since fpu is soft S-continuous, implies that f−1pu ((H,B)) and f−1pu ((K,B)) are soft semi-open in SS(X)B. As fpu is soft sur- jective, therefore (H,B) ˜6=Ỹ follows f−1pu ((H,B)) ˜6=X̃. Therefore f−1pu ((G,B))=̃f−1pu ((H,B))\̃f−1pu ((K,B)) is soft semi-D set. Hence the proof. S. Hussain / Eur. J. Pure Appl. Math, 10 (2) (2017), 199-210 208 Theorem 11. Let (X, τ,A) and (Y, τ∗, B) be soft topological spaces over X and Y respec- tively. Then the following statements are equivalent: (1) For any two distinct soft points eF ,eG in X̃A, there exists soft surjective soft S- continuous function fpu : SS(X)A → SS(Y )B, where Ỹ is soft semi-D1 space with fpu(eF ) ˜6=fpu(eG). (2) X̃ is soft semi-D1 space. Proof. (1) ⇒ (2) Since for any two distinct soft points eF , eG in X̃A, there exists soft surjective soft S-continuous fpu : SS(X)A → SS(Y )B, where Ỹ is soft semi-D1 space with fpu(eF ) ˜6=fpu(eG). Then there exist soft disjoint soft semi-D sets (G,B) and (H,B) in Ỹ with fpu(eF )∈̃(G,B), fpu(eG)∈̃(H,B). As fpu is soft surjective soft S-continuous, so Theorem 10 follows that f−1pu ((G,B)) and f−1pu ((H,B)) are soft disjoint soft semi-D sets in X̃ with eF ∈̃f−1pu ((G,B)), eG∈̃f−1pu ((H,B)). Hence again Theorem 10 implies that X̃ is soft semi-D1 space. (2) ⇒ (1) This follows by letting the identity soft function, which fulfills the desired properties. Hence the proof. Theorem 12. Let (X, τ,A) and (Y, τ∗, B) be soft topological spaces over X and Y respec- tively and soft function fpu : SS(X)A → SS(Y )B is soft bijective soft S-continuous. If Ỹ is soft semi-D1 space then X̃ is soft semi-D1 space. Proof. Suppose eF and eG be two distinct soft points in X̃A. Since fpu is soft injective and Ỹ is soft semi-D1, then there exist soft semi-D sets (G,B) and (H,B) in SS(Y )B such that fpu(eF )∈̃(G,B), fpu(eG)∈̃(H,B) and fpu(eG) /̃∈(G,B), fpu(eF ) /̃∈(H,B). There- fore, by Theorem 10, f−1pu ((G,B)) and f−1pu (H,B) are soft semi-D sets in SS(X)A with eF ∈̃f−1pu ((G,B)) and eG∈̃f−1pu (H,B). This implies that X̃ is soft semi-D1 space. This completes the proof. Theorem 13. Let (X, τ,A) and (Y, τ∗, B) be soft topological spaces over X and Y respec- tively. A soft function fpu : SS(X)A → SS(Y )B is soft S-continuous, if for each soft point eF in X̃A and each soft semi-open set (G,B) in SS(Y )B such that fpu(eF )∈̃(G,B), there exists a soft semi-open set (F,A) in SS(X)A such that fpu(F,A)⊆̃(G,B). Proof. (⇒) Since fpu is soft S-continuous, implies that f−1pu (G,B) is soft semi-open in SS(X)A, for soft semi-open set (G,B) in SS(Y )B. We need to show that there exists a soft semi-open set (F,A) in SS(X)A such that fpu(F,A)⊆̃(G,B), for each soft point eF in X̃A and each soft semi-open set (G,B) in SS(Y )B such that fpu(eF )∈̃(G,B). Consider the soft point eF in X̃A with eF ∈̃f−1pu (G,B) and (F,A) = f−1pu (G,B). This implies that for soft semi-open set (G,B), eF ∈̃(F,A) and fpu(F,A)⊆̃fpuf−1pu (G,B)⊆̃(G,B). (⇐) Suppose that for each soft point eF in X̃A and each soft semi-open set (G,B) in SS(Y )B such that f(eF )∈̃(G,B), there exists a soft semi-open set (F,A) in SS(X)A such that fpu(F,A)⊆̃(G,B). To prove that soft function fpu is soft S-continuous. We REFERENCES 209 show that the inverse image of soft semi-open set in SS(Y )B is soft semi-open set in SS(X)A. Now eF ∈̃f−1pu (G,B) follows fpu(eF )∈̃(G,B). Thus by hypothesis, there exists a soft semi-open set (F,A)eF such that eF ∈̃(F,A)eF and fpu((F,A)eF )∈̃(G,B). 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