@FMI c Kyung Moon Sa Co.

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1 Annals of Fuzzy Mathematics and Informatics Volume x, No. x, (Month 201y), pp. 1 xx ISSN: (print version) ISSN: (electronic version) c Kyung Moon Sa Co. Soft points and the structure of soft topological spaces Ningxin Xie Received 25 October 2014; Revised 12 December 2014; Accepted 25 February 2015 Abstract. Soft set theory is a new mathematical tool to deal with uncertain problems. Although soft sets are so-called sets, we can not handle them like ordinary sets. The reason is that soft sets are defined by mappings and they lack points. In this paper, the concept of soft points is introduced and the relationship between soft points and soft sets is considered. We prove the fact that soft sets can be translated into soft point sets and then may conveniently deal with soft sets as same as ordinary sets. Moreover, we investigate some soft point sets, define neighborhoods of soft points and reveal the structure of soft topological spaces such as interior, closure, boundary and bases by using neighborhoods of soft points AMS Classification: 49J53; 54A40. Keywords: Bases. Corresponding Author: Soft sets; Soft points; Soft point sets; Soft topologies; Neighborhoods; Ningxin Xie (ningxinxie100@126.com) 1. Introduction Most of traditional methods for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, many practical problems within fields such as economics, engineering, environmental science, medical science and social sciences involve data that contain uncertainties. We can not use traditional methods because of various types of uncertainties present in these problems. There are several theories: probability theory, theory of fuzzy sets [22], theory of interval mathematics, and theory of rough sets [17], which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties [16]. For example, theory of probabilities can deal only with stochastically stable phenomena. To overcome these kinds of difficulties, Molodtsov [16] proposed a completely new approach, which is called theory of soft sets, for modeling uncertainty.

2 Presently, works on soft set theory are progressing rapidly. Maji et al. [13, 15, 14] further studied soft set theory and used this theory to solve some decision making problems. Roy et al. [19] presented a fuzzy soft set theoretic approach towards decision making problems. Jiang et al. [11] generalized the adjustable approach to fuzzy soft sets based decision making. Jiang et al. [10] extended soft sets with description logics. Aktas and Cağman [2] defined soft groups. Feng et al. [6, 7] investigated the relationship among soft sets, rough sets and fuzzy sets. Ge et al. [8] discussed the relationship between soft sets and topological spaces. Babitha et al. [4] proposed relations on soft sets. Shabir et al. [21] introduced soft topological spaces over the universe with a fixed set of parameters. Cağman et al. [5] defined soft topologies on soft sets. Sen [20] investigated algebraic structure of soft sets. Although soft sets are so-called sets, we can not handle them like an ordinary set. The reason is that soft sets are defined by mappings and they lack points. Therefore, we need to introduce the concept of points in soft sets. The purpose of this paper is to introduce the concept of soft points. We prove that soft sets can be translated into soft point sets and then may conveniently deal with soft sets as same as ordinary sets. The rest of this paper is organized as follows. In Section 2, we give an overview of soft sets. In Section 3, we recall soft topological spaces. In Section 4, the concept of soft points is introduced, some soft point sets are studied and the relationship between soft points and soft sets is considered. We prove the fact that soft sets can be translated into soft point sets. In Section 5, we illustrate that soft relations and soft operations can be conveniently handled. Neighborhoods of soft points are proposed and the structure of soft topological spaces such as interior, closure, boundary and bases is revealed by using them. Conclusions are in Section Overview of soft sets In this section, we recall some basic concepts of soft sets. Throughout this paper, U denotes an initial universe, E denotes the set of parameters, 2 U denotes the power set of U. We only consider the case where U and E are both nonempty finite sets. Definition 2.1 ([16]). Let A E. A pair (f, A) is called a soft set over U, if f is a mapping given by f : A 2 U. We denote (f, A) by f A. In other words, a soft set over U is the parameterized family of subsets of the universe U. For ε A, f(ε) may be considered as the set of ε-approximate elements of f A. Obviously, a soft set is not a set. To illustrate this idea, let us consider the following Example 2.2. Example 2.2. Let U = {h 1, h 2, h 3, h 4, h 5, h 6 } be a set of houses and let A = {ε 1, ε 2, ε 3, ε 4 } E be a set of status of houses which stand for the parameters beautiful, modern, cheap and in the green surroundings, respectively. Now, we consider a soft set f A, which describes the attractiveness of the houses that Mr.X is going to buy. In this case, to define the soft set f A means to point out beautiful houses, modern houses and so on. Consider the mapping f given by houses(.), where (.) is to be filled in by one of the parameters ε i A. For instance, f(ε 1 ) means houses(beautiful), and its 2

3 Table 1. Tabular representation of the soft set f A h 1 h 2 h 3 h 4 h 5 h 6 ε ε ε ε functional value is the set consisting of all the beautiful houses in U. Let f(ε 1 ) = {h 1, h 2, h 5 }, f(ε 2 ) =, f(ε 3 ) = {h 3, h 4 }, f(ε 4 ) = {h 3, h 4, h 6 }. Then a soft set f A is described as the following Table 1. If h i f(ε j ), then h ij = 1; otherwise h ij = 0, where h ij are the entries in the table (see Table 1). Let A E. Denote S(U, A) = {f A : f A is a soft set over U}. Definition 2.3. Let A E and let f A S(U, A). We define (f A ) E S(U, E) by { f(ε) if ε A, f A (ε) =, if ε E A. Obviously, (f E ) E = f E. Definition 2.4 ([1]). Let A, B E and let f A S(U, A) and g B S(U, B). (1) f A is called a soft subset of g B, if A B and for any ε A, f(ε) g(ε). We write f A g B. (2) f A is called a soft super set of g B, if g B f A. We write f A g B. (3) f A and g B are called soft equal, if A = B and f(ε) = g(ε) for any ε A. We write f A = g B. Obviously, f A = g B if and only if f A g B and f A g B. Definition 2.5 ([13]). Let A, B E and let f A S(U, A) and g B S(U, B). (1) h A B is called the union of f A and g B, if f(ε) if ε A B, h(ε) = g(ε) if ε B A, f(ε) g(ε) if ε A B. We write f A g B = h A B. (2) h A B is called the intersection of f A and g B, if h(ε) = f(ε) g(ε) for any ε A B. We write f A g B = h A B. Remark 2.6. Let A, B, C E and let f A S(U, A), g B S(U, B) and h C S(U, C). Then (1) f A g B f A ( or g B ) f A g B. (2) If h C f A and h C g B, then h C f A g B. (3) If h C f A and h C g B, then h C f A g B. 3

4 Definition 2.7 ([21]). Let A E and let f A, g A, h A S(U, A). h A is called the difference of f A and g A, if h(ε) = f(ε) g(ε) for any ε A. We write h A = f A g A. Definition 2.8 ([13]). Let A E and let f A, g A S(U, A). g A is called the relative complement of f A, if g(ε) = U f(ε) for any ε A. We write g A = f A or (f A). Proposition 2.9 ([3]). Let A E and let f A, g A S(U, A). Then (1) (f A g A ) = f A g A. (2) (f A g A ) = f A g A. Remark Let A E and let f A, g A S(U, A). Then (1) (f A ) = f A. (2) f A g A (f A ) (g A ). Definition 2.11 ([21]). Let A E and let X U. The soft set X A over U is defined by X(ε) = X for any ε A. In this paper, U E and E are also denoted by Ũ and, respectively. Remark Let f A, g A S(U, A). Then (1) U A f A = f A, (2) f A g A = A f A g A, (3) f A g A = f A g A. Proposition Let A, B E and let f A S(U, A) and g B S(U, B). Then (1) (f A g B ) E = (f A ) E (g B ) E. (2) (f A g B ) E = (f A ) E (g B ) E. (3) (f A g A ) E = (f A ) E (g A ) E. Proof. (1) Denote h C = f A g B with C = A B. We will show that h C (ε) = f A (ε) g B (ε) for any ε E. Case 1 If ε A B, then ε C and ε E B. Since h C (ε) = h(ε) = f(ε), f A (ε) = f(ε) and g B (ε) =, we have h C (ε) = f A (ε) g B (ε). Case 2 If e B A, then ε C and ε E A. Since h C (ε) = h(ε) = g(ε), f A (ε) = and g B (ε) = g(ε), we have h C (ε) = f A (ε) g B (ε). Case 3 If ε A B, then ε C. Since h C (ε) = h(ε) = f(ε) g(ε), f A (ε) = f(ε) and g B (ε) = g(ε), we have h C (ε) = f A (ε) g B (ε). Case 4 If ε E A B, then ε (E A) (E B) (E C). Since h C (ε) =, f A (ε) = and g B (ε) =, we have h C (ε) = f A (ε) g B (ε). Thus, h C (ε) = f A (ε) g B (ε) for any ε E. This implies that (h C ) E = (f A ) E (g B ) E. Hence (f A g B ) E = (f A ) E (g B ) E. (2) The proof is similar to (1). (3) The proof is similar to (1). 3. Soft topological spaces In this section, we recall some concepts of soft topological spaces. 4

5 Definition 3.1 ([21]). τ S(U, E) is called a soft topology over U, if (i), Ũ τ; (ii) the union of any number of soft sets in τ belongs to τ; (iii) the intersection of any two soft sets in τ belongs to τ. The triplet (U, τ, E) is called a soft topological space over U. Every element of τ is called soft open set in U and its relative complement is called soft closed set in U. Definition 3.2 ([21]). Let (U, τ, E) be a soft topological space over U. f E S(U, E), the soft closure of f E is defined by cl(f E ) = {g E : f E g E and g E is a soft closed set in U}. For any Proposition 3.3 ([21]). Let (U, τ, E) be a soft topological space over U and let f E S(U, E). Then (1) cl( ) = and cl(ũ) = Ũ. (2) f E cl(f E ). (3) f E is a soft closed set in U f E = cl(f E ). (4) cl(cl(f E )) = cl(f E ). Definition 3.4 ([9]). Let (U, τ, E) be a soft topological space over U. f E S(U, E), the soft interior of f E is defined by int(f E ) = {g E : g E f E and g E τ}. For any Proposition 3.5 ([9]). Let (U, τ, E) be a soft topological space over U. Then for any f E S(U, E), int(f E ) = Ũ cl(ũ f E). Proposition 3.6 ([9]). Let (U, τ, E) be a soft topological space over U and let f E S(U, E). Then (1) int( ) = and int(ũ) = Ũ. (2) int(f E ) f E. (3) f E τ f E = int(f E ). (4) int(int(f E )) = int(f E ). Definition 3.7 ([9]). Let (U, τ, E) be a soft topological space over U. f E S(U, E), the soft boundary of f E is defined by (f E ) = cl(f E ) cl(f E). 4. Soft points For any In this section, we will introduce the concept of soft points and consider the relationship between soft points and soft sets The concept of soft points. In this subsection, we define soft points, which originate from the concept of fuzzy points (see [12, 18]). Definition 4.1. Let f E S(U, E). f E is called a soft point over U, if there exist e E and x U such that { {x}, if ε = e, f(ε) =, if ε E {e}. 5

6 We denote f E by (x e ) E. In this case, x is called the support point of (x e ) E, {x} is called the support set of (x e ) E and e is called the expressive parameter of (x e ) E. The family of all soft points over U is denoted by P (U, E). Let A E and let f A S(U, A). We denote P (U, A) = {(x e ) E P (U, E) : e A}, F(A) = {(x e ) E : x f(e) and e A}. Remark 4.2. (1) (x e ) E F(A) x f(e) and e A. (2) F(A) = f(e). e A Example 4.3. Let U = {x 1, x 2, x 3, x 4, x 5 } and E = {e 1, e 2, e 3, e 4 }. We define f(e 1 ) = {x 1, x 4 }, f(e 2 ) = U, f(e 3 ) = {x 5 }, f(e 4 ) =. Then F(E) = {((x 1 ) e1 ) E, ((x 4 ) e1 ) E, ((x 1 ) e2 ) E, ((x 2 ) e2 ) E, ((x 3 ) e2 ) E, ((x 4 ) e2 ) E, ((x 5 ) e2 ) E, ((x 5 ) e3 ) E } and P (U, E) = {((x i ) ej ) E : 1 i 5, 1 j 4}. Proposition 4.4. Let A, B E. Then (1) A B = P (U, A) P (U, B). (2) P (U, E) = {P (U, A) : A E}. Proof. These are obvious. Proposition 4.5. Let f A, g A, h A S(U, A). (1) If g A f A, then G(A) F(A). (2) If f A = g A h A, then F(A) = G(A) H(A). (3) If f A = g A h A, then F(A) = G(A) H(A). (4) If f A = g A h A, then F(A) = G(A) H(A). Proof. (1) This is obvious. (2) Let (x e ) E F(A). Then x f(e). Since f A = g A h A, we have x g(e) and x h(e). Thus (x e ) E G(A) and (x e ) E F(A). Hence (x e ) E G(A) H(A). Conversely. The proof is similar. (3) The proof is similar to (2). (4) The proof is similar to (2). Proposition 4.6. (1) If f A = U A, then P (U, A) = F(A). (2) P (U, A) = {F(A) : f A S(U, A)}. Proof. (1) This is obvious. (2) Let f A S(U, A). Since f A U A, by Proposition 4.5 and (1) we have F(A) P (U, A). Thus P (U, A) {F(A) : f A S(U, A)}. Conversely, since U A S(U, A), by (1) we have P (U, A) {F(A) : f A S(U, A)}. Hence P (U, A) = {F(A) : f A S(U, A)}. 6

7 4.2. Soft points and soft sets. In this subsection, we will investigate the relationship between soft points and soft sets. Definition 4.7. Let A E and let f A S(U, A) and (x e ) E P (U, E). We define (x e ) E f A by (x e ) E (f A ) E. Note that (x e ) E fa, if (x e ) E (fa ) E. Remark 4.8. (1) (x e ) E = (x e ) E x = x and e = e. (2) (x e ) E f A x f(e) and e A (x e ) E F(A). (3) (x e ) E f A and f A g B = (x e ) E g B. Theorem 4.9. Let f A S(U, A) and let (x e ) E P (U, A). Then (x e ) E f A if and only if (x e ) E f A. Proof. Let (x e ) E f A. Then x f(e) and e A. This implies that x U f(e) = f (e). By Remark 4.8, (x e ) E f A. Conversely. Let (x e ) E f A. Since (x e ) E P (U, A), e A. By Remark 4.8, x f (e). Thus x f(e). This show that (x e ) E f A. Theorem Let f A S(U, A). Then f A = F(A). Proof. Denote h A = F(A). Then h A = {(x e ) E : x f(e) and e A}. Thus For any ε A, we have h(ε) = x e (ε) = ( e A x f(e) x f(ε) h A = e A x f(e) x e (e)) ( (x e ) E. ε A {e} x f(e) This show that h A = f A. Hence f A = F(A). x e (ε)) = ( x f(ε) {x}) = f(ε). Remark Theorem 4.10 reveal the fact that soft sets can be translated into soft point sets. 5. Some related results with soft points In this section, we obtain some related results with soft points Soft relations and soft operations. In this subsection, we will prove that soft relations and soft operations can be respectively translated into ordinary relations and ordinary operations, and then may conveniently deal with them. Theorem 5.1. Let A, B E and let f A S(U, A) and g B S(U, B). Then (1) f A g B A B and F(A) G(B). (2) f A = g B A = B and F(A) = G(B). 7

8 Proof. (1) Necessity. Let (x e ) E F(A). By Proposition 4.6, (x e ) E P (U, A). Since A B, by Proposition 4.4, we have P (U, A) P (U, B). Thus (x e ) E P (U, B) and so (x e ) E G(B). Hence F(A) G(B). Sufficiency. Let A B. For any e A and x f(e), we have (x e ) E F(A). Since F(A) G(B), (x e ) E G(B). Then x g(e) and e B. Thus f(e) g(e) and f A g B. (2) This holds by (1). Theorem 5.2. Let A, B, C E and let f A S(U, A), g B S(U, B) and h C S(U, C). (1) If C = A B, then h C = f A g B H(C) = F(A) G(B). (2) If C = A B, then h C = f A g B H(C) = F(A) G(B). (3) f A = g A h A F(A) = G(A) H(A). Proof. (1) Necessity. Let (x e ) E F(A) G(B). Then x f(e) and e A, or x g(e) and e B. Case 1 If e A B, then x f(e) = h(e) and e C. Thus (x e ) E H(C). Case 2 If e B A, then x g(e) = h(e) and e C. Thus (x e ) E H(C). Case 3 If e A B, then x f(e) g(e) = h(e) and e C. Thus (x e ) E H(C). Hence F(A) G(B) H(C). Conversely. Let (x e ) E H(C). Then x h(e) and e C. Case 1: If e A B, then x h(e) = f(e) and e A. Thus (x e ) E F(A) and so (x e ) E F(A) G(B). Case 2: If e B A, then x h(e) = g(e) and e B. Thus (x e ) E G(B) and so (x e ) E G(B) F(A). Case 3: If e A B, then x h(e) = f(e) g(e). So x f(e) and e A, or x g(e) and e B. Thus (x e ) E F(A) G(B). Hence H(C) F(A) G(B). Therefore, F(A) G(B) = H(C). Sufficiency. Denote l C = f A g B. By the proof of Necessity, L(C) = F(A) G(B). Since F(A) G(B) = H(C), L(C) = H(C). By Theorem 5.1, h C = l C. Thus f A g B = h A B. (2) The proof is similar to (1). (3) This holds by Proposition Neighborhoods of soft points. In this subsection, we define neighborhoods of soft points. Definition 5.3. Let (U, τ, E) be a soft topological space over U. Let f E S(U, E) and (x e ) E P (U, E). Then (1) f E is called a neighborhood of (x e ) E, if there exists g E τ such that (x e ) E g E f E. (2) f E is called an open neighborhood of (x e ) E, if f E τ and f E is a neighborhood of (x e ) E. The family of all neighborhoods of (x e ) E is denoted by N (xe) E. Denote τ (xe) E = {f E τ : (x e ) E f E }. Then τ (xe) E means the family of all open neighborhoods of (x e ) E. 8

9 Example 5.4. Let U = {x 1, x 2, x 3 }, E = {e 1, e 2 }. We define g 1 (e 1 ) = {x 2 }, g 1 (e 2 ) = {x 3 }; g 2 (e 1 ) = {x 2 }, g 2 (e 2 ) = {x 1, x 3 }; g 3 (e 1 ) = {x 2, x 3 }, g 3 (e 2 ) = {x 3 }; g 4 (e 1 ) = {x 2, x 3 }, g 4 (e 2 ) = {x 1, x 3 }; g 5 (e 1 ) = {x 1, x 2 }, g 5 (e 2 ) = {x 2, x 3 }; g 6 (e 1 ) = {x 1, x 2 }, g 6 (e 2 ) = U; g 7 (e 1 ) = U, g 7 (e 2 ) = {x 2, x 3 }; f(e 1 ) = {x 1, x 2 }, f(e 2 ) = {x 1, x 3 }. Put τ = {, Ũ, (g 1) E, (g 2 ) E,, (g 7 ) E }. Then (U, τ, E) is a soft topological space over U. Obviously, (g 2 ) E f E τ. Since x 1 g 2 (e 2 ), ((x 1 ) e2 ) E (g 2 ) E. Then ((x 1 ) e2 ) E (g 2 ) E f E. So f E is a neighborhood of ((x 1 ) e2 ) E. But f E is not an open neighborhood of ((x 1 ) e2 ) E. Since x 1 g 4 (e 2 ) x 1 g 6 (e 2 ), ((x 1 ) e2 ) E (g 4 ) E τ, ((x 1 ) e2 ) E (g 6 ) E τ. Thus τ ((x1) e2 ) E = {(g 2 ) E, (g 4 ) E, (g 6 ) E, Ũ}. Proposition 5.5. Let (U, τ, E) be a soft topological space over U and let N (xe) E is a neighborhood system of (x e ) E. Then (1) If f E N (xe) E, then (x e ) E f E ; (2) If f E, g E N (xe) E, then f E g E N (xe) E ; (3) If g E f E and g E N (xe) E, then f E N (xe) E ; Proof. (1) Let f E N (xe) E. Then there exists g E τ such that (x e ) E g E f E. Thus x g(e) f(e). By Remark 4.8, (x e ) E f E. (2) Let f E, g E N (xe) E. Then there exist h E, k E τ such that (x e ) E h E f E and (x e ) E k E g E. So x h(e) k(e) and h E k E f E g E. By Remark 4.8, (x e ) E h E k E. Now h E k E τ. Thus f E g E N (xe) E. (3) Let g E N (xe) E. Then there exists h E τ such that (x e ) E h E g E. Since g E f E, (x e ) E h E f E. Thus f E N (xe) E. Theorem 5.6. Let (U, τ, E) be a soft topological space over U and let f E S(U, E). Then the following are equivalent. (1) f E τ; (2) f E τ (xe) E for any (x e ) E f E ; (3) f E N (xe) E for any (x e ) E f E. Proof. (1) (2) and (2) (3) are obvious. (3) (1). Let f E N (xe) E for any (x e ) E f E. Then there exists g(x e ) E τ such that (x e ) E g(x e ) E f E. So (x e ) E g(x e ) E f E. This implies that {(x e ) E : x f(e) and e E} {g(x e ) E : x f(e) and e E} f E By Theorem 4.10, f E = {(x e ) E : x f(e) and e E}. Then f E = {g(x e ) E : x f(e) and e E}. Hence f E τ. Corollary 5.7. Let (U, τ, E) be a soft topological space over U. Then Proof. This holds by Theorem 5.6. τ = {f E : f E N (xe) E for any (x e ) E f E }. 9

10 5.3. The structure on soft topological spaces. In this subsection, we will reveal the structure on soft topological spaces such as interior, closure, boundary and bases by using soft points and their neighborhoods. Definition 5.8. Let (U, τ, E) be a soft topological space over U. Let f E S(U, E) and (x e ) E P (U, E). (1) (x e ) E is called an interior point of f E, if g E f E for some g E N (xe) E. (2) (x e ) E is called a adherent point of f E, if g E f E for any g E N (xe) E. (3) (x e ) E is called a boundary point of f E, if g E f E and g E f E for any g E N (xe) E. Remark 5.9. (x e ) E is a boundary point of f E (x e ) E is a adherent point of f E and (x e ) E is a adherent point of f E. Proposition Let (U, τ, E) be a soft topological space over U. Let f E S(U, E) and (x e ) E P (U, E). (1) (x e ) E is an interior point of f E, if g E f E for some g E τ (xe) E. (2) (x e ) E is a adherent point of f E, if g E f E for any g E τ (xe) E. (3) (x e ) E is a boundary point of f E, if g E f E and g E f E for any g E τ (xe) E. Proof. The proof is straightforward. Theorem Let (U, τ, E) be a soft topological space over U. f E S(U, E) int(f E ) = {(x e ) E : (x e ) E is an interior point of f E }. Proof. Denote h E = int(f E ). We will show that H(E) = {(x e ) E : (x e ) E is an interior point of f E }. Then for any Let (x e ) E H(E). Then x h(e). Since h(e) = {g(e) : g E f E and g E τ}, there exists g E τ such that g E f E and x g(e). x g(e) implies that (x e ) E g E. Then f E N (xe) E. Thus (x e ) E is an interior point of f E. This show that H(E) {(x e ) E : (x e ) E is an interior point of f E }. Conversely. Let (x e ) E is an interior point of f E. By Proposition 5.10, g E f E for some g E τ (xe) E. (x e ) E g E implies that x g(e). Then (x e ) E G(E). Note that g E τ and g E f E. By Remark 2.6, g E int(f E ) = h E. By Proposition 4.5, G(E) H(E). Hence (x e ) E H(E). This show that H(E) {(x e ) E : (x e ) E is an interior point of f E }. Hence H(E) = {(x e ) E : (x e ) E is an interior point of f E }. By Theorem 4.10, h E = H(E). Therefore, int(f E ) = {(x e ) E : (x e ) E is an interior point of f E }. Theorem Let (U, τ, E) be a soft topological space over U. f E S(U, E) cl(f E ) = {(x e ) E : (x e ) E is a adherent point of f E }. Proof. Denote h E = cl(f E ). We will show that H(E) = {(x e ) E : (x e ) E is a adherent point of f E }. 10 Then for any

11 Let (x e ) E H(E). Suppose that f E g E = for some g E τ (xe) E. Since f E g E =, by Remark 2.12, we have f E g E. Note that h E = {l E : f E l E and l E τ}. By Remark 2.6, h E g E. Since (x e) E H(E), x h(e). Then x g (e). This implies that x g(e). Since g E τ (xe) E, (x e ) E g E. Thus x g(e), a contradiction. Hence f E g E for any g E τ (xe) E. By Proposition 5.10, (x e ) E is a adherent point of f E. This show that H(E) {(x e ) E : (x e ) E is a adherent point of f E }. Conversely. Let (x e ) E is a adherent point of f E. Suppose that (x e ) E H(E). Then x h(e). By Remark 4.8, (x e ) E he. By Theorem 4.9, (x e ) E h E. By Proposition 3.3, h E τ and f E h E = (h E ). By Remark 2.12, f E h E =. Since (x e ) E is a adherent point of f E, by Proposition 5.10, we have f E g E for any g E τ (xe) E. Now h E τ (x e) E. This implies that f E h E, a contradiction. This show that H(E) {(x e ) E : (x e ) E is a adherent point of f E }. Hence H(E) = {(x e ) E : (x e ) E is a adherent point of f E }. By Theorem 4.10, h E = H(E). Therefore, cl(f E ) = {(x e ) E : (x e ) E is a adherent point of f E }. Theorem Let (U, τ, E) be a soft topological space over U. f E S(U, E) (f E ) = {(x e ) E : (x e ) E is a boundary point of f E }. Then for any Proof. Denote h E = (f E ), g E = cl(f E ) and l E = cl(f E ). Since h E = g E l E, by Proposition 4.5, we have H(E) = G(E) L(E). By the proof of Theorem 5.12, G(E) = {(x e ) E : (x e ) E is a adherent point of f E } and L(E) = {(x e ) E : (x e ) E is a adherent point of f E }. Thus H(E) = {(x e ) E : (x e ) E is both a adherent point of f E and f E}. Hence H(E) = {(x e ) E : (x e ) E is a boundary point of f E }. By Theorem 4.10, h E = H(E). Therefore, (f E ) = {(x e ) E : (x e ) E is a boundary point of f E }. Definition Let (U, τ, E) be a soft topological space over U and let B τ. B is called a base for (U, τ, E), if for any f E τ, there exists B B such that f E = {b E : b E B }. Example Let U = {x 1, x 2, x 3 } and E = {e 1, e 2 }. We define f 1 (e 1 ) =, f 1 (e 2 ) = {x 1 }; f 2 (e 1 ) =, f 2 (e 2 ) = {x 2 }; f 3 (e 1 ) =, f 3 (e 2 ) = {x 1, x 2 }; f 4 (e 1 ) = {x 1 }, f 4 (e 2 ) = ; f 5 (e 1 ) = {x 2 }, f 5 (e 2 ) = ; f 6 (e 1 ) = {x 1, x 2 }, f 6 (e 2 ) = ; f 7 (e 1 ) = {x 1 }, f 7 (e 2 ) = {x 2 }; f 8 (e 1 ) = {x 2 }, f 8 (e 2 ) = {x 1 }; f 9 (e 1 ) = {x 2 }, f 9 (e 2 ) = {x 2 }; f 10 (e 1 ) = {x 2 }, f 10 (e 2 ) = {x 1, x 2 }; f 11 (e 1 ) = {x 1, x 2 }, f 11 (e 2 ) = {x 2 }; f 12 (e 1 ) = {x 1, x 3 }, f 12 (e 2 ) = {x 1, x 3 }; f 13 (e 1 ) = U, f 13 (e 2 ) = {x 1, x 3 }; f 14 (e 1 ) = {x 1, x 3 }, f 14 (e 2 ) = U; f 15 (e 1 ) = {x 1, x 2 }, f 15 (e 2 ) = {x 1 }; f 16 (e 1 ) = {x 1 }, f 16 (e 2 ) = {x 1 }; f 17 (e 1 ) = {x 1 }, f 17 (e 2 ) = {x 1, x 2 }; f 18 (e 1 ) = {x 1, x 2 }, f 18 (e 2 ) = {x 1, x 2 }. Put τ = {, Ũ, (f 1) E, (f 2 ) E,, (f 18 ) E }. Then (U, τ, E) is a soft topological space over U. 11

12 Put B = {, (f 1 ) E, (f 2 ) E, (f 4 ) E, (f 5 ) E, (f 9 ) E, (f 14 ) E, (f 15 ) E, (f 17 ) E }. Then B is a base for (U, τ, E). Theorem Let (U, τ, E) be a soft topological space over U and let f E S(U, E) and let B be a base for (U, τ, E). Then f E τ (x e ) E F(E) b E B such that (x e ) E b E f E. Proof. Necessity. Suppose that f E τ. Let (x e ) E F(E). Since B is base, there exists B B such that f E = {b E : b E B }. By Remark 4.8, x f(e) = {b(e) : b E B for any e E}. Then there exists b E B such that x b(e). By Remark 4.8, (x e ) E b E. Note that b E f E. Hence (x e ) E b E f E. Sufficiency. Let x f(e) and e E. Then (x e ) E F(E). So there exists b(e, x) E B such that (x e ) E b(e, x) E f E. So (x e ) E b(e, x) E f E. This implies that {(x e ) E : x f(e) and e E} {b(e, x) E : x f(e) and e E} f E By Theorem 4.10, f E = {(x e ) E : x f(e) and e E}. Then f E = {b(e, x) E : x f(e) and e E}. Note that B τ. Thus f E τ. Corollary Let (U, τ, E) be a soft topological space over U and let B be a base for (U, τ, E). Then τ = {f E S(U, E) : (x e ) E f E, b E B such that (x e ) E b E f E }. Proof. This holds by Theorem Theorem Let (U, τ, E) be a soft topological space over U and let B τ. Then the following are equivalent. (1) B is base; (2) For any (x e ) E P (U, E) and f E τ (xe) E, there exists b E B such that (x e ) E b E f E ; (3) For any (x e ) E P (U, E) and f E N (xe) E, there exists b E B such that (x e ) E b E f E. Proof. (1) (2). Let (x e ) E P (U, E) and let f E τ (xe) E. Then there exists g E τ such that (x e ) E g E f E. By (1) there exists B B such that g E = {b E : b E B }. Thus (x e ) E g E. By Remark 4.8, x g(e) = {b(e) : b E B for any e E}. Then there exists b E B such that x b(e). By Remark 4.8, (x e ) E b E. Hence b E g E f E and b E f E. (2) (3). This is obvious. (3) (1). Let f E τ. Let x f(e) and e E. By Remark 4.8, (x e ) E f E. Thus f E N (xe) E. By (2) there exists b(e, x) E B such that (x e ) E b(e, x) E f E. So (x e ) E b(e, x) E f E. This implies that {(x e ) E : x f(e) and e E} {b(e, x) E : x f(e) and e E} f E By Theorem 4.10, f E = {(x e ) E : x f(e) and e E}. Then f E = {b(e, x) E : x f(e) and e E}. Put B = {b(e, x) E : x f(e) and e E}. Then B is base. 12

13 Let (U, τ, E) be a soft topological space over U and let X U. Denote τ X = {g E X E : g E τ}. It is easy to prove that (X, τ X, E) is a soft topological space over X. (X, τ X, E) is called a subspace of (U, τ, E). Theorem 5.19 ([21]). Let (U, τ, E) be a soft topological space over U, let (X, τ X, E) be its subspace and let g E X E. Then (1) g E is soft open in X if and only if g E = f E X E for some f E τ. (2) g E is soft closed in X if and only if g E = f E X E for some soft closed set f E in U. Theorem Let (U, τ, E) be a soft topological space over U, let B be a base for (U, τ, E) and let X U. Then B X = {f E X E : f E B} is a base for (X, τ X, E). Proof. Let g E τ X. By Theorem 5.19, there exists f E τ such that g E = f E X E. Let x f(e) and e E. By Remark 4.8, (x e ) E f E. Thus f E N (xe) E. By Theorem 5.18, there exists b(e, x) E B such that (x e ) E b(e, x) E f E. This implies that (x e ) E b(e, x) E X E f E X E = g E. Note that b(e, x) E X E B X. By Theorem 5.18, B X is a base for (X, τ X, E). 6. Conclusions In this paper, we defined soft points, researched some soft point sets and presented neighborhoods of soft points. It was worthwhile to mention that soft sets can be translated into soft point sets and then we may conveniently deal with soft relations, soft operations and soft topological spaces. In future work, we will study continuous mappings between soft topological spaces by using soft points and their neighborhoods. Acknowledgements. This work is supported by the National Natural Science Foundation of China ( ), the Natural Science Foundation of Guangxi (2014GXNSFAA118001), the Science Research Project of Guangxi University for Nationalities (2012MDZD036), the Science Research Project 2014 of the China- ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities (KT201427). References [1] A.Aygünoǧlu, H.Aygün, Introduction to fuzzy soft groups, Comput. Math. Appl. 58(2009) [2] H.Aktas, N.Cağman, Soft sets and soft groups, Inform. Sci. 177(2007) [3] M.I.Ali, F.Feng, X.Y.Liu, W.K.Min, M.Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57(2009) [4] K.V.Babitha, J.J.Sunil, Soft set relations and functions, Comput. Math. Appl. 60(2010) [5] N.Cağman, S.Karatas, S.Enginoglu, Soft topology, Comput. Math. Appl. 62(2011) [6] F.Feng, C.Li, B.Davvaz, M.Irfan Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14(2010) [7] F.Feng, X.Liu, V.Leoreanu-Fotea, Y.Jun, Soft sets and soft rough sets, Inform. Sci. 181(2011) [8] X.Ge, Z.Li, Y.Ge, Topological spaces and soft sets, J. Comput. Anal. Appl. 13(2011)

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