Hindawi Publishing Corporation ξ e Scientiο¬c World Journal Volume 2014, Article ID 843456, 6 pages http://dx.doi.org/10.1155/2014/843456
Research Article On Soft π½-Open Sets and Soft π½-Continuous Functions Metin Akdag and Alkan Ozkan Department of Mathematics, Science Faculty, Cumhuriyet University, Sivas, Turkey Correspondence should be addressed to Alkan Ozkan; alkan
[email protected] Received 19 February 2014; Accepted 27 May 2014; Published 17 June 2014 Academic Editor: Naim Cagman Copyright Β© 2014 M. Akdag and A. Ozkan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the concepts soft π½-interior and soft π½-closure of a soft set in soft topological spaces. We also study soft π½-continuous functions and discuss their relations with soft continuous and other weaker forms of soft continuous functions.
1. Introduction and Preliminary The concept of soft sets was first introduced by Molodtsov [1] in 1999 who began to develop the basics of corresponding theory as a new approach to modeling uncertainties. In [1, 2], Molodtsov successfully applied the soft theory in several directions such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, and theory of measurement. In recent years, an increasing number of papers have been written about soft sets theory and its applications in various fields [3, 4]. Shabir and Naz [5] introduced the notion of soft topological spaces which are defined to be over an initial universe with a fixed set of parameters. In addition, Maji et al. [6] proposed several operations on soft sets, and some basic properties of these operations have been revealed so far. In general topology, the concept of π½-open sets was introduced in [7] and π½-open sets have been referred to as semipreopen by Andrijevic [8]. In [9] this concept has been generalized to soft setting. Our motivation in this paper is to define soft π½-interiors and soft π½-closures and investigate their properties which are important for further research on soft topology. These researches not only can form the theoretical basis for further applications of topology on soft sets but also lead to the development of information system and various fields in engineering. Furthermore, we will study soft π½-continuous functions and obtain some characterizations of such functions. Definition 1 (see [1]). Let π be an initial universe and let πΈ be a set of parameters. Let π(π) denote the power set of π and
let π΄ be a nonempty subset of πΈ. A pair (πΉ, π΄) is called a soft set over π, where πΉ is a mapping given by πΉ : π΄ β π(π). In other words, a soft set over π is a parameterized family of subsets of the universe π. For π β π΄, πΉ(π) may be considered as the set of π-approximate elements of the soft set (πΉ, π΄). Definition 2 (see [6]). A soft set (πΉ, π΄) over π is called a null soft set, denoted by Ξ¦, if π β π΄, πΉ(π) = 0. Definition 3 (see [6]). A soft set (πΉ, π΄) over π is called an Μ if π β π΄, πΉ(π) = π. absolute soft set, denoted by π΄, If π΄ = πΈ, then the π΄-universal soft set is called a universal Μ soft set, denoted by π. Definition 4 (see [5]). Let π be a nonempty subset of π; then Μ denotes the soft set (π, πΈ) over π for which π(π) = π, for π all π β πΈ. Definition 5 (see [6]). The union of two soft sets of (πΉ, π΄) and (πΊ, π΅) over the common universe π is the soft set (π», πΆ), where πΆ = π΄ βͺ π΅ and for all π β πΆ, πΉ (π) , if π β π΄ β π΅, { { π» (π) = {πΊ (π) , if π β π΅ β π΄, { πΉ βͺ πΊ , if π β π΄ β© π΅. (π) (π) {
(1)
Μ (πΊ, π΅) = (π», πΆ). We write (πΉ, π΄)βͺ Definition 6 (see [6]). The intersection (π», πΆ) of two soft sets (πΉ, π΄) and (πΊ, π΅) over a common universe π, denoted by
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Μ (πΊ, π΅), is defined as πΆ = π΄ β© π΅, and π»(π) = πΉ(π) β© πΊ(π) (πΉ, π΄)β© for all π β πΆ. Definition 7 (see [6]). Let (πΉ, π΄) and (πΊ, π΅) be two soft sets Μ (πΊ, π΅), if π΄ β π΅, and over a common universe π. (πΉ, π΄)β π»(π) = πΉ(π) β πΊ(π) for all π β π΄. Definition 8 (see [5]). Let π be the collection of soft sets over π; then π is said to be a soft topology on π if it satisfies the following axioms: Μ belong to π, (1) Ξ¦, π (2) the union of any number of soft sets in π belongs to π, (3) the intersection of any two soft sets in π belongs to π. The triplet (π, π, πΈ) is called a soft topological space over π. Let (π, π, πΈ) be a soft topological space over π; then the members of π are said to be soft open sets in π. The relative complement of a soft set (πΉ, π΄) is denoted by (πΉ, π΄)π and is defined by (πΉ, π΄)π = (πΉπ , π΄) where πΉπ : π΄ β π(π) is a mapping given by πΉπ (π) = π β πΉ(π) for all π β π΄. Let (π, π, πΈ) be a soft topological space over π. A soft set (πΉ, π΄) over π is said to be a soft closed set in π, if its relative complement (πΉ, π΄)π belongs to π. If (π, π, πΈ) is a soft topological space with Μ then π is called the soft indiscrete topology on π π = {Ξ¦, π}, and (π, π, πΈ) is said to be a soft indiscrete topological space. If (π, π, πΈ) is a soft topological space with π being the collection of all soft sets which can be defined over π, then π is called the soft discrete topology on π and (π, π, πΈ) is said to be a soft discrete topological space. Definition 9. Let (π, π, πΈ) be a soft topological space over π and let (πΉ, π΄) be a soft set over π. (1) Reference [4]: the soft interior of (πΉ, π΄) is the soft set int((πΉ, π΄)) = βͺ{(π, π΄) : (π, π΄) is soft open and Μ (πΉ, π΄)}. (π, π΄)β (2) Reference [5]: the soft closure of (πΉ, π΄) is the soft set cl((πΉ, π΄)) = β©{(πΉ, πΈ) : (πΉ, πΈ) is soft closed and Μ (πΉ, πΈ)}. (πΉ, π΄)β Clearly cl((πΉ, π΄)) is the smallest soft closed set over π which contains (πΉ, π΄) and int((πΉ, π΄)) is the largest soft open set over π which is contained in (πΉ, π΄). Definition 10. A soft set (πΉ, π΄) of a soft topological space (π, π, πΈ) is said to be (a) soft open [5] if its complement is soft closed, Μ int(cl(int((πΉ, π΄)))), (b) soft πΌ-open [10] if (πΉ, π΄)β Μ int(cl((πΉ, π΄))), (c) soft preopen [9] if (πΉ, π΄)β Μ cl(int((πΉ, π΄))), (d) soft semiopen [11] if (πΉ, π΄)β Μ cl(int(cl((πΉ, π΄)))). (e) soft π½-open [9] if (πΉ, π΄)β Proposition 11. (a) Every soft open set is soft πΌ-closed. (b) Every soft πΌ-open set is soft preopen. (c) Every soft πΌ-open set is soft semiopen. (d) Every soft semiopen set is soft π½-open. (e) Every soft preclosed set is soft π½-open.
Soft open set Soft πΌ-open set
Soft semi-open set
Soft pre-open set Soft π½-open set
Figure 1
Proof. The proof is obvious from Definition 10. Remark 12. We have following implications; however, the converses of these implications are not true, in general, as shown in Figure 1. Example 13. Let π = {π₯1 , π₯2 , π₯3 }, πΈ = {π1 , π2 , π3 }, and Μ (πΉ1 , πΈ), (πΉ2 , πΈ), (πΉ3 , πΈ), ..., (πΉ15 , πΈ)}, where (πΉ1 , πΈ), π = {Ξ¦, π, (πΉ2 , πΈ), (πΉ3 , πΈ), ...(πΉ15 , πΈ) are soft sets over π, defined as follows: (πΉ1 , πΈ) = {(π1 , {π₯1 , π₯2 }), (π2 , {π₯3 }), (π3 , {π₯1 , π₯3 })}, (πΉ2 , πΈ) = {(π1 , {π₯2 }), (π2 , {π₯1 , π₯2 }), (π3 , {π₯1 , π₯2 })}, (πΉ3 , πΈ) = {(π1 , {π₯2 }), (π3 , {π₯1 })}, Μ (π3 , π)}, Μ (πΉ4 , πΈ) = {(π1 , {π₯1 , π₯2 }), (π2 , π), (πΉ5 , πΈ) = {(π1 , {π₯3 }), (π2 , {π₯1 , π₯3 }), (π3 , {π₯2 })}, (πΉ6 , πΈ) = {(π2 , {π₯3 })}, Μ (π2 , {π₯1 , π₯3 }), (π3 , π)}, Μ (πΉ7 , πΈ) = {(π1 , π), (πΉ8 , πΈ) = {(π2 , {π₯1 }), (π3 , {π₯2 })}, Μ (π3 , {π₯1 , π₯2 })}, (πΉ9 , πΈ) = {(π1 , {π₯2 , π₯3 }), (π2 , π), (πΉ10 , πΈ) = {(π1 , {π₯2 , π₯3 }), (π2 , {π₯1 , π₯3 }), (π3 , {π₯1 , π₯2 })}, (πΉ11 , πΈ) = {(π2 , {π₯1 , π₯3 }), (π3 , {π₯2 })}, Μ (π3 , {π₯1 , π₯2 })}, (πΉ12 , πΈ) = {(π1 , {π₯2 }), (π2 , π), (πΉ13 , πΈ) = {(π1 , {π₯2 }), (π2 , {π₯1 }), (π3 , {π₯1 , π₯3 })}, Μ (πΉ14 , πΈ) = {(π1 , {π₯1 , π₯2 }), (π2 , {π₯1 , π₯3 }), (π3 , π)}, (πΉ15 , πΈ) = {(π1 , {π₯2 }), (π2 , {π₯3 }), (π3 , {π₯1 })}. Then π defines a soft topology on π, and thus (π, π, πΈ) is a soft topological space over π. Clearly the soft closed sets are Μ Ξ¦, (πΉ1 , πΈ)π , (πΉ2 , πΈ)π , (πΉ3 , πΈ)π , ..., (πΉ15 , πΈ)π . π, Then, let us take (πΉ, πΈ) = {(π1 , {π₯1 , π₯3 }), (π2 , {π₯2 }), Μ int(cl(int((πΉ, πΈ)))) = π, Μ (π3 , {π₯1 , π₯2 })}; then int(πΉ, πΈ) = π, Μ int(cl(int((πΉ, πΈ)))); hence, (πΉ, πΈ) is soft πΌ-open and so (πΉ, πΈ)β set but not soft open set (since (πΉ, πΈ) is not soft open set). Now, let us take (πΊ, πΈ) = {(π1 , {π₯2 , π₯3 }), (π2 , {π₯3 }), (π3 , {π₯1 })}; then int((πΊ, πΈ)) = {(π1 , {π₯2 }), (π2 , {π₯3 }), (π3 , {π₯1 })}, Μ (π2 , {π₯2 , π₯3 }), (π2 , {π₯1 , π₯3 })}, and so cl(int((πΊ, πΈ))) = {(π1 , π), Μ cl(int((πΊ, πΈ))); hence, (πΊ, πΈ) is soft semiopen set but (πΊ, πΈ)β not soft πΌ-open set. Now, let us take (πΏ, πΈ) = {(π1 , {π₯2 }), (π2 , {π₯3 })}; then cl((πΏ, πΈ)) = (πΉ8 , πΈ)π , int(cl((πΏ, πΈ))) = {(π1 , {π₯1 , π₯2 }), (π2 , {π₯3 }), Μ int(cl((πΏ, πΈ))); hence, (πΏ, πΈ) is (π3 , {π₯1 , π₯3 })}, and so (πΏ, πΈ)β soft preopen set but not soft πΌ-open set.
The Scientific World Journal Μ (π2 , {π₯2 }), Finally, let us consider (π», πΈ) = {(π1 , π), (π3 , {π₯1 , π₯3 })} as a soft set in π. Μ and so (π», πΈ)β Μ cl(int(cl(π», Then cl(int(cl(π», πΈ))) = π, πΈ))); as a result, (π», πΈ) is soft π½-open set, but it is neither soft semiopen set nor soft preopen set.
2. Some Properties of Soft π½-Open Sets and Soft π½-Closed Sets Recall that a soft set (πΉ, π΄) of a soft topological space (π, π, πΈ) Μ cl(int(cl(πΉ, π΄))). The is said to be soft π½-open [9] if (πΉ, π΄)β complement of a soft π½-open set is called soft π½-closed. Soft π½closure and soft π½-interior of a soft set are defined as follows. Definition 14. Let (π, π, πΈ) be a soft topological space and let (πΉ, π΄) be a soft set over π. (a) Soft π½-interior of a soft set (πΉ, π΄) in π is denoted by Μ {(π, π΄) : (π, π΄) is a soft π½-open set π π½ int((πΉ, π΄)) = βͺ Μ (πΉ, π΄)}. and (π, π΄)β (b) Soft π½-closure of a soft set (πΉ, π΄) in π is denoted Μ {(πΉ, πΈ) : (πΉ, πΈ) is a soft π½-closed by π π½ cl((πΉ, π΄)) = β© Μ (πΉ, πΈ)}. set and (πΉ, π΄)β Clearly π π½ cl((πΉ, π΄)) is the smallest soft π½-closed set over π which contains (πΉ, π΄) and π π½ int((πΉ, π΄)) is the largest soft π½-open set over π which is contained in (πΉ, π΄). We will denote the family of all soft π½-open sets (resp., soft π½-closed sets) of a soft topological space (π, π, πΈ) by ππ½ππ(π, π, πΈ) (resp., ππ½πΆπ(π, π, πΈ)). Proposition 15 (see [12]). (1) Arbitrary union of soft π½-open sets is a soft π½-open set. (2) Arbitrary intersection of soft π½closed sets is a soft π½-closed set. Proposition 16. Let (π, π, πΈ) be a soft topological space and let (πΉ, π΄) be a soft set over π; then (1) (πΉ, π΄) β ππ½πΆπ(π, π, πΈ) β (πΉ, π΄) = π π½ cl((πΉ, π΄)); (2) (πΉ, π΄) β ππ½ππ(π, π, πΈ) β (πΉ, π΄) = π π½ int((πΉ, π΄)). Μ {(πΉ, πΈ) : (πΉ, πΈ) is a Proof. (1) Let (πΉ, π΄) = π π½ cl((πΉ, π΄)) = β© Μ (πΉ, πΈ)}. This shows that (πΉ, π΄) β soft π½-closed set and (πΉ, π΄)β Μ (πΉ, πΈ)}. {(πΉ, πΈ) : (πΉ, πΈ) is a soft π½-closed set and (πΉ, π΄)β Hence (πΉ, π΄) is soft π½-closed set. Conversely, let (πΉ, π΄) be soft π½-closed set. Μ (πΉ, π΄) and (πΉ, π΄) is a soft π½-closed set, Since (πΉ, π΄)β Μ (πΉ, π΄) β {(πΉ, πΈ) : (πΉ, πΈ) is a soft π½-closed set and (πΉ, π΄)β (πΉ, πΈ)}. Μ (πΉ, πΈ) for all such (πΉ, πΈ)βs. Further, (πΉ, π΄)β Μ {(πΉ, πΈ) : (πΉ, πΈ) is a soft π½-closed set and (πΉ, π΄)β Μ (πΉ, π΄) = β© (πΉ, πΈ)}. (2) Similar to (1). Proposition 17. In a soft space (π, π, πΈ), the following hold for soft π½-closure: (1) π π½ cl(Ξ¦) = Ξ¦. (2) π π½ cl((πΉ, π΄)) is soft π½-closed set in (π, π, πΈ) for each soft subset (πΉ, π΄) of π. Μ π π½ cl((πΊ, π΅)), if (πΉ, π΄)β Μ (πΊ, π΅). (3) π π½ cl((πΉ, π΄))β
3 Theorem 18. Let (π, π, πΈ) be a soft topological space and let (πΉ, π΄) and (πΊ, π΅) be two soft sets over π; then (1) (π π½ cl((πΉ, π΄)))π = π π½ int((πΉ, π΄)π ); (2) (π π½ int((πΉ, π΄)))π = π π½ cl((πΉ, π΄)π ); Μ (πΊ, π΅) β π π½ int((πΉ, π΄))β Μ π π½ int((πΊ, π΅)); (3) (πΉ, π΄)β Μ = π; Μ (4) π π½ cl(Ξ¦) = Ξ¦ and π π½ cl(π) Μ = π; Μ (5) π π½ int(Ξ¦) = Ξ¦ and π π½ int(π) Μ (πΊ, π΅))β Μ π π½ cl((πΉ, π΄))βͺ Μ π π½ cl((πΊ, π΅)); (6) π π½ cl((πΉ, π΄)βͺ Μ (πΊ, π΅))β Μ π π½ int((πΉ, π΄))β© Μ π π½ int((πΊ, π΅)); (7) π π½ int((πΉ, π΄)β© (8) π π½ cl(π π½ cl((πΉ, π΄))) = π π½ cl((πΉ, π΄)); (9) π π½ int(π π½ int((πΉ, π΄))) = π π½ int((πΉ, π΄)). Proof. Let (πΉ, π΄) and (πΊ, π΅) be two soft sets over π. Μ {(πΉ, π΄) : (πΉ, π΄)β Μ (πΉ, π΄) (1) We have (π π½ cl((πΉ, π΄)))π = (β© Μ {(πΉ, π΄)π : (πΉ, π΄)β Μ and (πΉ, π΄) β ππ½πΆπ(π, π, πΈ)})π = βͺ Μ {(πΉ, π΄)π : (πΉ, π΄) and (πΉ, π΄) β ππ½πΆπ(π, π, πΈ)} = βͺ Μ (πΉ, π΄)π and (πΉ, π΄)π β ππ½ππ(π, π, πΈ)} = (πΉ, π΄)π β π π½ int((πΉ, π΄)π ). (2) Similar to (1). (3) Follows from definition. Μ are soft π½-closed sets so π π½ cl(Ξ¦) = Ξ¦ (4) Since Ξ¦ and π Μ = π. Μ and π π½ cl(π) Μ are soft π½-open sets so π π½ int(Ξ¦) = Ξ¦ (5) Since Ξ¦ and π Μ Μ and π π½ int(π) = π. Μ ((πΉ, π΄)βͺ Μ (πΊ, π΅)) and (πΊ, π΅)β Μ ((πΉ, π΄)βͺ Μ (6) We have (πΉ, π΄)β Μ (πΊ, π΅)). Then by Proposition 17 (3), π π½ cl((πΉ, π΄))β Μ (πΊ, π΅)) and cl((πΊ, π΅))β Μ π π½ cl((πΉ, π΄)βͺ Μ (πΊ, π π½ cl((πΉ, π΄)βͺ Μ π π½ cl((πΉ, π΄))β Μ π π½ cl((πΉ, π΄)βͺ Μ (πΊ, π΅)). π΅)) β π π½ cl((πΊ, π΅))βͺ (7) Similar to (6). (8) Since π π½ cl((πΉ, π΄)) β ππ½πΆπ(π, π, πΈ), so by Proposition 16 (1), π π½ cl(π π½ cl((πΉ, π΄))) = π π½ cl((πΉ, π΄)). (9) Since π π½ int((πΉ, π΄)) β ππ½ππ(π, π, πΈ), so by Proposition 16 (2), π π½ int(π π½ int((πΉ, π΄))) = π π½ int((πΉ, π΄)).
Theorem 19. For a soft topological space (π, π, πΈ) the following are valid. Μ ππ½ππ(π, π, πΈ). (a) πβ (b) If (πΉ, π΄) is a soft set in π and (πΊ, π΅) is a soft preopen Μ (πΉ, π΄)β Μ cl(int((πΊ, π΅)), then set in π such that (πΊ, π΅)β (πΉ, π΄) is a soft π½-open set. Proof. (a) The proof is obvious. (b) Since (πΊ, π΅) is a soft Μ int(cl((πΊ, π΅))). preopen set we have (πΊ, π΅)β Μ cl(int(int(cl((πΊ, π΅))))) = cl(int(cl((πΊ, π΅))))β Μ Then (πΉ, π΄)β cl(int(cl((πΉ, π΄)))), so (πΉ, π΄) is a soft π½-open set. Definition 20 (see [13]). Let (π, π, πΈ) be soft topological space and let π be an ordinary subset of π. Then ππ = ((πΉ, π΄)/π : (πΉ, π΄) β π) is a soft topology on π and is called the induced or
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relative soft topology. The pair (π, ππ ) is called a soft subspace of (π, π, πΈ): (π, ππ) is called a soft open/soft closed/soft π½open soft subspace if the characteristic function of π, namely, ππ , is soft open/soft closed/soft π½-open, respectively. Theorem 21. Let (π, π, πΈ) be a soft topological space. Suppose Μ πβ Μ π and (π, ππ ) is a soft π½-open soft subspace of (π, π, πΈ). πβ Then π is soft π½-open soft subspace in π if and only if π is soft π½-open soft subspace in π. Proof. Suppose that π is soft π½-open soft subspace in π. Μ cl(int(cl(ππ))). Then ππ β Μ ππ = ππ so that ππ β© Μ ππ = Μ π implies ππ β© But πβ Μ cl(int(cl(ππ ))) = ππ β© Μ ππ . ππ β This implies that ππ is soft π½-open in π. That is, π is soft π½-open soft subspace in π.
3. Soft π½-Continuity Definition 22 (see [14]). Let (π, πΈ) and (π, πΎ) be soft classes. Let π’ : π β π and π : πΈ β πΎ be mappings. Then a mapping π : (π, πΈ) β (π, πΎ) is defined as follows: for a soft set (πΉ, π΄) in (π, πΈ), (π(πΉ, π΄), π΅), π΅ = π(π΄) β πΎ is a soft set in (π, πΎ) given by π(πΉ, π΄)(π½) = π’(βͺπΉ(πΌ)πΌβπβ1 (π½)β©π΄ ) for π½ β πΎ. (π(πΉ, π΄), π΅) is called a soft image of a soft set (πΉ, π΄). If π΅ = πΎ, then we will write (π(πΉ, π΄), πΎ) as π(πΉ, π΄). Definition 23 (see [14]). Let π : (π, πΈ) β (π, πΎ) be a mapping from a soft class (π, πΈ) to another soft class (π, πΎ), and let (πΊ, πΆ) be a soft set in soft class (π, πΎ), where πΆ β πΎ. Let π’ : π β π and π : πΈ β πΎ be mappings. Then (πβ1 (πΊ, πΆ), π·), π· = πβ1 (πΆ), is a soft set in the soft classes (π, πΈ), defined as follows: πβ1 (πΊ, πΆ)(πΌ) = π’β1 (πΊ(π(πΌ))) for πΌ β π· β πΈ. (πβ1 (πΊ, πΆ), π·) is called a soft inverse image of (πΊ, πΆ). Hereafter we will write (πβ1 (πΊ, πΆ), πΈ) as πβ1 (πΊ, πΆ). Theorem 24 (see [14]). Let π : (π, πΈ) β (π, πΎ), π’ : π β π, and π : πΈ β πΎ be mappings. Then for soft sets (πΉ, π΄), (πΊ, π΅) and a family of soft sets (πΉπ , π΄ π ) in the soft class (π, πΈ), we have the following: (1) π(Ξ¦) = Ξ¦, Μ = π, Μ (2) π(π) Μ (πΊ, π΅)) = π(πΉ, π΄)βͺ Μ π(πΊ, π΅) in general (3) π((πΉ, π΄)βͺ Μ Μ π(βͺπ (πΉπ , π΄ π )) = βͺπ π(πΉπ , π΄ π ), Μ π(πΉ, π΄)β© Μ (πΊ, π΅))β Μ π(πΊ, π΅) (4) π((πΉ, π΄)β© in general Μβ© Μ π π(πΉπ , π΄ π ), Μ π (πΉπ , π΄ π ))β π(β© Μ (πΊ, π΅), then π(πΉ, π΄)β Μ π(πΊ, π΅), (5) if (πΉ, π΄)β (6) πβ1 (Ξ¦) = Ξ¦, Μ = π, Μ (7) πβ1 (π) Μ (πΊ, π΅)) = πβ1 (πΉ, π΄)βͺ Μ πβ1 (πΊ, π΅) in general (8) πβ1 ((πΉ, π΄)βͺ Μ π (πΉπ , π΄ π )) = βͺ Μ π πβ1 (πΉπ , π΄ π ), πβ1 (βͺ Μ (πΊ, π΅)) = πβ1 (πΉ, π΄)β© Μ πβ1 (πΊ, π΅) in general (9) πβ1 ((πΉ, π΄)β© β1 Μ β1 Μ π π (πΉπ , π΄ π ), π (β©π (πΉπ , π΄ π )) = β© Μ πβ1 (πΊ, π΅). Μ (πΊ, π΅), then πβ1 (πΉ, π΄)β (10) if (πΉ, π΄)β
Soft continity Soft πΌ-continity
Soft semi-continity
Soft pre-continity
Soft π½-continity
Figure 2
Throughout the paper, the spaces π and π stand for soft topological spaces with ((π, π, πΈ) and (π, V, πΎ)) assumed unless otherwise stated. Moreover, throughout this paper, a soft mapping π : π β π stands for a mapping, where π : (π, π, πΈ) β (π, π, πΎ), π’ : π β π, and π : πΈ β πΎ are assumed mappings unless otherwise stated. Definition 25. A soft mapping π : π β π is called soft π½-continuous [12] (resp., soft πΌ-continuous [10], soft precontinuous [10], and soft semicontinuous [15]) if the inverse image of each soft open set in π is soft π½-open (resp., soft πΌ-open, soft preopen, and soft semiopen) set in π. Remark 26. We have the following implications; however, the converses of these implications are not true, in general, as shown in Figure 2. Example 27. Let π = {π₯1 , π₯2 , π₯3 }, π = {π¦1 , π¦2 , π¦3 }, πΈ = {π1 , π2 , π3 }, and πΎ = {π1 , π2 , π3 } and let (π, π, πΈ) and (π, π, πΎ) be soft topological spaces. Define π’ : π β π and π : πΈ β πΎ as π’(π₯1 ) = {π¦1 }, π’(π₯2 ) = {π¦3 }, π’(π₯3 ) = {π¦2 }, π(π1 ) = {π2 }, π(π2 ) = {π1 }, π(π3 ) = {π3 }. Let us consider the soft topology π on π given in Example 13; that is, Μ (πΉ1 , πΈ), (πΉ2 , πΈ), (πΉ3 , πΈ), ..., (πΉ15 , πΈ)}, π = π = {Ξ¦, π, Μ (πΉ, πΎ)}; {Ξ¦, π, (πΉ, πΎ) = {(π1 , {π¦1 , π¦2 }), (π2 , {π¦3 }), (π3 , {π¦1 , π¦3 })} and mapping; π : (π, π, πΈ) β (π, π, πΎ) is a soft mapping. Then (πΉ, πΎ) is a soft open set in π; πβ1 ((πΉ, πΎ)) = {(π1 , {π₯1 , π₯3 }), (π2 , {π₯2 }), (π3 , {π₯1 , π₯2 })} is a soft πΌ-open set but not soft open set in π. Therefore, π is a soft πΌ-continuous function but not soft continuous function. Example 28. Let π = {π₯1 , π₯2 , π₯3 }, π = {π¦1 , π¦2 , π¦3 }, πΈ = {π1 , π2 , π3 }, and πΎ = {π1 , π2 , π3 } and let (π, π, πΈ) and (π, π, πΎ) be soft topological spaces. Let us consider the π’ : π β π and π : πΈ β πΎ as mapping given in Example 27 and the soft topology π on π given in Example 13; that is, Μ (πΊ, πΎ)}, (πΊ, πΎ) = {(π1 , {π¦3 }), (π2 , {π¦2 })} and π = {Ξ¦, π, mapping;
The Scientific World Journal π : (π, π, πΈ) β (π, π, πΎ) is a soft mapping. Then (πΊ, πΎ) is a soft open set in π; πβ1 ((πΊ, πΎ)) = {(π1 , {π₯2 }), (π2 , {π₯3 })} is a soft preopen set but not soft πΌ-open set in π. Thus, π is a soft precontinuous function but not soft πΌ-continuous function. Example 29. Let π = {π₯1 , π₯2 , π₯3 }, π = {π¦1 , π¦2 , π¦3 }, πΈ = {π1 , π2 , π3 }, and πΎ = {π1 , π2 , π3 } and let (π, π, πΈ) and (π, π, πΎ) be soft topological spaces. Let us consider the π’ : π β π and π : πΈ β πΎ as mapping given in Example 27 and the soft topology π on π given in Example 13; that is, Μ (πΏ, πΎ)}, (πΏ, πΎ) = {(π1 , {π¦3 , π¦2 }, (π2 , {π¦2 }), π = {Ξ¦, π, (π3 , {π¦1 })} and mapping; π : (π, π, πΈ) β (π, π, πΎ) is a soft mapping. Then (πΏ, πΎ) is a soft open set in π; πβ1 ((πΏ, πΎ)) = {(π1 , {π₯2 , π₯3 }), (π2 , {π₯3 }), (π3 , {π₯1 })} is a soft semiopen set but not soft πΌ-open set in π. Hence, π is a soft semicontinuous function but not soft πΌcontinuous function. Example 30. Let π = {π₯1 , π₯2 , π₯3 }, π = {π¦1 , π¦2 , π¦3 }, πΈ = {π1 , π2 , π3 }, and πΎ = {π1 , π2 , π3 } and let (π, π, πΈ) and (π, π, πΎ) be soft topological spaces. Let us consider the π’ : π β π and π : πΈ β πΎ as mapping given in Example 27 and the soft topology π on π given in Example 13; that is, Μ (π2 , {π¦3 }), Μ (π», πΎ)}, (π», πΎ) = {(π1 , π), π = {Ξ¦, π, (π3 , {π¦1 , π¦2 })} and mapping; π : (π, π, πΈ) β (π, π, πΎ) is a soft mapping. Then (π», πΎ) is a soft open set in π; Μ (π2 , {π₯2 }), (π3 , {π₯1 , π₯3 })} is a πβ1 ((π», πΎ)) = {(π1 , π), soft π½-open set, but it is neither soft semiopen set nor soft preopen set in π. Therefore, π is a soft π½-continuous function, but it is neither soft semicontinuous function nor soft precontinuous function. Definition 31 (see [12]). Let π : π β π be a function. π is called soft π½-irresolute if the inverse image of soft π½-open set in π is soft π½-open in π. Definition 32. Let π : π β π be a function. π is called soft π½-open if the image of each soft π½-open set in π is soft π½-open in π. Theorem 33. Let π : (π, π, πΈ) β (π, πσΈ , πΈ) be a soft continuous and soft open set. Then π is soft π½-open set. Μ Proof. Let (πΉ, π΄) be any soft π½-open set. Then (πΉ, π΄)β cl(int(cl((πΉ, π΄)))). Therefore, Μ π(cl(int(cl((πΉ, π΄)))))β Μ cl(int(cl(π((πΉ, π΄))))), π((πΉ, π΄))β β π((πΉ, π΄)) is soft π½-open.
5 This shows that π is soft π½-open set. Theorem 34. If (πΉ, π΄) is soft closed and (πΊ, π΅) is soft π½-open Μ (πΊ, π΅) is soft π½-open. then (πΉ, π΄)βͺ Μ cl(int(cl((πΊ, π΅)))). Now Proof. By hypothesis (πΊ, π΅)β Μ (πΊ, π΅)β Μ (πΉ, π΄)βͺ Μ cl(int(cl((πΊ, π΅)))) , (πΉ, π΄)βͺ Μ cl(int(cl((πΊ, π΅))))β Μ cl(int(cl(((πΉ, = cl(int(cl((πΉ, π΄))))βͺ Μ (πΊ, π΅))))). π΄)βͺ Μ (πΊ, π΅) is soft π½-open set. This shows that (πΉ, π΄)βͺ Theorem 35. Let π : (π, π, πΈ) β (π, πσΈ , πΈ) be soft continuous and soft open. Then π is soft π½-irresolute. Proof. Let (πΉ, π΄) be any soft π½-open set in π. Then Μ cl(int(cl((πΉ, π΄)))). Since π is soft continuous and soft (πΉ, π΄)β open it follows that Μ πβ1 (cl(int(cl((πΉ, π΄))))), πβ1 ((πΉ, π΄))β Μ cl(int(πβ1 (cl((πΉ, π΄))))), = cl(πβ1 (int(cl((πΉ, π΄))))) β = cl(int(cl(πβ1 ((πΉ, π΄))))). This shows that πβ1 ((πΉ, π΄)) is soft π½-open. This shows that π is soft π½-irresolute. Proposition 36. A function π : π β π is soft π½-irresolute if and only if for every soft π½-closed set (πΉ, πΎ) of π, πβ1 ((πΉ, πΎ)) is soft π½-closed. Proposition 37. In a soft topological space (π, π, πΈ) the following are valid: (a) (πΉ, π΄) is soft π½-openβ π π½ int((πΉ, π΄)) = (πΉ, π΄). (b) (πΉ, π΄) is soft π½-closedβ π π½ cl((πΉ, π΄)) = (πΉ, π΄). Theorem 38. π : (π, π, πΈ) β (π, πσΈ , πΈ) is soft π½-irresolute if and only if for every soft set (πΉ, π΄) of π, π(π π½ cl((πΉ, Μ π π½ cl(π((πΉ, π΄))). π΄)))β Proof. Suppose that π is soft π½-irresolute. Now π(π π½ cl((πΉ, π΄))) is soft π½-closed set. By hypothesis πβ1 (π π½ cl(π((πΉ, π΄)))) is soft π½-closed set. Μ πβ1 (π((πΉ, π΄)))β Μ πβ1 (π π½ cl(π((πΉ, π΄)))). Hence, And (πΉ, π΄)β Μ πβ1 (π π½ cl by the definition of soft π½-closure, π π½ cl((πΉ, π΄))β (π((πΉ, π΄)))). Μ π π½ cl(π((πΉ, π΄))). That is π(π π½ cl((πΉ, π΄)))β Conversely, suppose that (πΉ, π΄) is soft π½-closed set in Μ π π½ cl(π(πβ1 ((πΉ, π. Now by hypothesis π(π π½ cl(πβ1 ((πΉ, π΄))))β Μ πβ1 ((πΉ, π΄)) so π΄)))) = (πΉ, π΄). This implies π π½ cl(πβ1 ((πΉ, π΄)))β β1 β1 that π ((πΉ, π΄)) = π π½ cl(π ((πΉ, π΄))). That is πβ1 ((πΉ, π΄)) is soft π½-closed set and so π is soft π½irresolute. Theorem 39. π : π β π is soft π½-irresolute if and only if for Μ πβ1 (π π½ cl((πΉ, πΎ))). all soft sets (πΉ, πΎ) of π, π π½ cl(πβ1 ((πΉ, πΎ)))β Proof. Suppose π is soft π½-irresolute. Now π π½ cl((πΉ, πΎ)) is soft π½-closed set so that πβ1 (π π½ cl((πΉ, πΎ))) is soft π½Μ πβ1 (π π½ cl((πΉ, πΎ))), it follows closed set. Since πβ1 ((πΉ, πΎ))β
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Μ from the definition of soft π½-closure that π π½ cl(πβ1 ((πΉ, πΎ)))β β1 π (π π½ cl((πΉ, πΎ))). Conversely suppose that (πΉ, πΎ) is soft π½-closed set in π. Then π π½ cl((πΉ, πΎ)) = (πΉ, πΎ). Μ πβ1 (π π½ππ((πΉ, πΎ))) Now by hypothesis π π½ cl(πβ1 ((πΉ, πΎ)))β β1 = π ((πΉ, πΎ)). Therefore, π π½ cl(πβ1 ((πΉ, πΎ))) = πβ1 (π π½ cl((πΉ, πΎ))) = β1 π ((πΉ, πΎ)). Thus, πβ1 ((πΉ, πΎ)) is soft π½-closed set and so π is soft π½irresolute. The following results are easy to establish. Proposition 40. Suppose π : π β π and π : π β π are both soft π½-irresolute. Then πππ : π β π is soft π½-irresolute. Proposition 41. Let π : π β π be soft continuous and soft open. Then (a) π is soft π½-irresolute; (b) πβ1 (π π½ cl((πΉ, πΎ))) = π π½ cl(πβ1 (((πΉ, πΎ)))), with (πΉ, πΎ) being a soft set in π. Definition 42. Let π and π be soft topological spaces. π and π are said to be M-soft π½-homeomorphic if and only if there exists π : π β π such that π is 1-1, onto, M soft π½-continuous and soft π½-open. Such an π is called soft π½-homeomorphism. Proposition 43. If π : π β π is soft π½-homeomorphism, then πβ1 (π π½ cl((πΉ, πΎ))) = π π½ cl(πβ1 ((πΉ, πΎ))), where (πΉ, πΎ) is a soft set in π. Corollary 44. If π : π β π is a soft π½-homeomorphism, then (a) π π½ cl(π((πΉ, πΎ))) = π(π π½ cl((πΉ, πΎ))), (b) π(π π½ int((πΊ, π΅))) = π π½ int(π((πΊ, π΅))), (c) πβ1 (π π½ int((πΉ, πΎ))) = π π½ int(πβ1 ((πΉ, πΎ))).
4. Conclusion In this paper, we introduce the concept of soft π½-interior and soft π½-closure of a soft set in topological spaces and study some of their properties. We also introduce the concept of soft π½-open sets and soft π½-continuous functions in topological spaces and some of their properties have been established. We hope that the findings in this paper are just the beginning of a new structure and not only will form the theoretical basis for further applications of topology on soft sets but also will lead to the development of information system and various fields in engineering.
Conflict of Interests There is no conflict of interests regarding the publication of this paper.
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