Applications of Soft Dense Sets to Soft Continuity

Advances in Mathematics: Scientific Journal 9 (2020), no.6, 3623–3630 ADV MATH ISSN: 1857-8365 (printed); 1857-8438 (electronic) SCI JOURNAL https://doi.org/10.37418/amsj.9.6.40 APPLICATIONS OF SOFT DENSE SETS TO SOFT CONTINUITY SANDEEP KAUR, ARJAN SINGH1 , AND NAVPREET SINGH NOORIE A BSTRACT. In this paper, we introduce the concept of soft dense subsets. We give characterizations of soft continuity in terms of soft dense subsets and in terms of convergent sequences of soft maps taken from soft dense subsets. We obtain (i) an extension theorem for a soft continuous map which is defined on soft dense subsets of the domain (ii) soft continuity of a soft map in terms of the soft continuity of its restrictions to members of a soft cover of the domain having soft dense intersection. 1. I NTRODUCTION Most of the complex problems in engineering, computer sciences, medical sciences, environment etc. have various uncertainties which can not be solved by classical methods. To describe and extract the useful information hidden in uncertain data, researchers in mathematics, computer science and related areas have proposed a number of theories such as intuitionistic fuzzy set theory, fuzzy set theory and rough set theory. But all these theories have their own limitations as pointed out in [6], the inadequacy of the parametrization tool of the theories is possibly, the reason for these limitations. To deal with such type of vagueness and uncertainties, Molodtsov in [6], introduced a new approach called soft set theory which are free from the difficulties affecting existing methods and started to develop the basics of the corresponding theory. There has been a rapid growth 1 corresponding author 2010 Mathematics Subject Classification. 54A05, 54C50, 54C60, 54C99. Key words and phrases. Soft metric spaces, Soft dense sets, Soft points, Soft continuous, Soft closed. 3623 3624 S. KAUR, A. SINGH, AND N. S. NOORIE of interest in soft set theory and its application in recent years. Xiao et al. [14] and Pei and Miao in [10] discussed the relationship between soft sets and information systems. A soft set can be represented by Boolean-valued information system as we may see the structure of a soft set can classify the objects into two classes (1 or 0) and so it can be used to represent a data set. The theory of soft sets has been applied to data analysis and decision support system. Shabir and Naz [12] initiated the study of soft topological space and later, a lot of work about soft topological spaces has been done by various authors in [12], [1], [5] etc. In these studies, the concept of soft points were studied in various forms. In this paper, we use the concept of soft points defined in [8]. Das and Samanta in [3], introduced the notion of soft real sets and soft real numbers. In [4], they introduced the concept of soft metric spaces by using the notion of soft points and investigated some of its basic properties. In [9], maps between metric spaces and dense subsets in metric spaces are studied. Also, some map gluing theorems which describe the continuity of maps between metric spaces in terms of dense subsets of the domain were obtained. Characterizations of continuity of a map between metric spaces, in terms of convergent sequences taken from dense subset of the domain of the map and in terms of restriction of the map to a dense subset of the domain are given. The above study of maps between metric spaces lead us to study of soft maps and soft dense subsets in soft metric spaces. In section 2, we recall the basic concept related to soft metric spaces. In section 3, we first define soft dense subsets and give characterization of soft continuous maps between soft metric spaces in terms of soft dense subsets of domain space [Theorem 3.2 below]. We also give characterization of soft continuity in terms of convergent sequence of soft points in soft dense subset of domain of the soft map [Theorem 3.5 below]. As corollaries to Theorem 3.5 we obtain (i) an extension theorem for a soft continuous map which is defined on soft dense subsets of the domain [Corollary 3.1 below], (ii) soft continuity of a soft map in terms of the soft continuity of its restrictions to members of a soft cover of the domain having soft dense intersection [Corollary 3.2]. APPLICATIONS OF SOFT DENSE SETS TO SOFT CONTINUITY 3625 2. P RELIMINARIES Zorlutuna, Min and Atmaca studied soft topological spaces in [13] and introduced some important notions related to soft sets which we shall use in our results in this paper. Let A be the set of parameters. A soft set F : A → P (X), over X is defined as a parameterized family of subsets of the universe X denoted by (F, A) or FA and a soft real set is a mapping F : A → B(R), where B(R) be the collection of all non-empty bounded subsets of real numbers R. Detailed study of soft real numbers and related concepts is given in [3] and [4] where the definition of soft continuity between soft metric spaces is defined in [7]. e A) will denote the absolute soft set and soft e and (X, e d, Throughout the paper, X metric space with soft metric de respectively. e A) is called soft sequential compact e d, Definition 2.1. [2] A soft metric space (X, e metric space if every soft sequence has a soft subsequence that converges in X. The following results will be utilized in this paper. Theorem 2.1. [4] Let (Ye , deY , A) be a soft metric subspace of a soft metric space e A) and P x ∈ e d, e e (Y, A). Then for any soft open ball SS(B(Pax , re)) in X, (X, a e (Y, A) is soft open ball in (Ye , deY , A), and also any soft open ball SS(B(Pax , re)) ∩ e with (Y, A). in (Ye , deY , A) is obtained as the intersection of a soft open ball in X Theorem 2.2. [7] A soft mapping (ϕ, e) is soft continuous at the soft point Pax e if and only if for every sequence of soft points {Paxn }n converging to the soft ∈X n e A), the sequence {(ϕ, e)(P xn )}n in (Ye , ρe, B) e d, point Pax in the soft metric space (X, an converges to a soft point (ϕ, e)(Pax ) in Ye . e A) be a soft metric space. A soft point P x ∈ e d, e is a eX Theorem 2.3. [11] Let (X, a soft limit point of (F,A) if and only if there is a soft sequence of soft points in (F,A) converging to Pax . 3. R ESULTS In this section, we study soft continuity of a soft map between soft metric spaces in terms of soft dense subsets of the domain space. We begin by introe Throughout this paper, ducing the following definition of soft dense subset of X. e (Z, A) will denote an arbitrarily fixed soft dense subset of X. 3626 S. KAUR, A. SINGH, AND N. S. NOORIE e will be called soft dense in X e if (Z, A) = X. e Definition 3.1. A soft set (Z, A) of X e and e eX e (Z, A) such e 0̄ there exists soft point Pby ∈ In other words, for every Pax ∈ ǫ> y x e ,P ) < ee that d(P ǫ. a b Our first theorem follows directly from Theorem 2.3 above: e is said to be soft dense in X e if and only if for Theorem 3.1. A soft set (Z, A) of X e there exists a sequence {Padn }n of soft points in (Z, A) converging to eX every Pax ∈ n Pax . Following theorem gives characterization of soft continuous maps in soft metric spaces in terms of soft dense subsets of domain space: e A) and (Ye , ρe, B) be two soft metric spaces. A soft mapping e d, Theorem 3.2. Let (X, e A) → (Ye , ρe, B) is soft continuous if and only if (ϕ, e)|(Z,A) is soft e d, (ϕ, e) : (X, continuous and (ϕ, e) is soft continuous at each point of (Z, A)c where (Z, A) is e soft dense in X. Proof. Let Pax ∈ (Z, A) and suppose (ϕ, e) is not soft continuous at Pax . Then e 0̄ and a sequence {Paxnn }n (ϕ, e)|(Z,A) is soft continuous implies there exists an e ǫ> ee in (Z, A)c such that Paxnn −→ Pax but ρe((ϕ, e)(Paxnn ), (ϕ, e)(Pax )) ≥ ǫ for every n. As x e e e e (Z, A) e 0̄ there exist Pby ∈ (Z, A) is soft dense in X, for every Pa ∈ X and e ǫ> y x e ,P ) < e e ǫ. Now since (ϕ, e) is soft continuous at each point of such that d(P a b xn e c e then for each n, there exists a soft point Pan ∈(Z, A) and (Z, A) is soft dense in X yn y e xn , P n ) < e (Z, A) such that d(P e n1̄ . By e n1̄ and ρe((ϕ, e)(Paxnn ), (ϕ, e)(Pbynn )) < Pb n ∈ an bn e xn , P yn ) < e n1̄ , we get Pbynn −→ Pax . Since (ϕ, e)|(Z,A) is soft conPaxnn −→ Pax and d(P an bn tinuous at Pax , it follows that (ϕ, e)(Pbynn ) −→ (ϕ, e)(Pax ) and so e n1̄ implies that (ϕ, e)(Paxnn ) −→ (ϕ, e)(Pax ). This conρe((ϕ, e)(Paxnn ), (ϕ, e)(Pbynn )) < e e tradicts our assumption that ρe((ϕ, e)(Paxnn ), (ϕ, e)(Pax )) ≥ ǫ for every n. Hence x (ϕ, e) is soft continuous at Pa and therefore, (ϕ, e) is soft continuous.  The proof of the following theorem, on soft continuity of soft maps between soft metric space, follows from above theorem and easily proved fact that if (ϕ, e)|(F,A) is soft continuous and (F, A) is soft open then (ϕ, e) is soft continuous at each soft point of (F, A): e A) and (Ye , ρe, B) be two soft metric spaces. Let (ϕ, e) : e d, Theorem 3.3. Let (X, e A) → (Ye , ρe, B) be a soft mapping and X e d, e = (F, A) ∪ e (G, A) where (F, A) is (X, APPLICATIONS OF SOFT DENSE SETS TO SOFT CONTINUITY 3627 e Then (ϕ, e) is soft continuous if (ϕ, e)|(F,A) soft open and (G, A) is soft dense in X. and (ϕ, e)|(G,A) are both soft continuous. The following theorem is a map gluing theorem on soft continuity of a soft map between soft metric spaces: e A) and (Ye , ρe, B) be two soft metric spaces. Let (ϕ, e) : e d, Theorem 3.4. Let (X, e A) → (Ye , ρe, B) be a soft mapping and X e d, e = (M, A) ∪ e (N, A) where (M, A) (X, e Then (ϕ, e) is soft continuous if (ϕ, e)|(M,A) and e (N, A) is soft dense in X. ∩ (ϕ, e)|(N,A) are both soft continuous. e Without loss of generality, we may assume that P x ∈ e X. e (M, A). Proof. Let Pax ∈ a e 0̄ and a sequence ǫ> Let (ϕ, e) is not soft continuous at Pax then there exist e xn xn x xn e e e {Pan }n in X such that Pan −→ Pa but ρe((ϕ, e)(Pan ), (ϕ, e)(Pax )) ≥ ǫ for evx xn ery n. Since, (ϕ, e)|(M,A) is soft continuous at Pa , this sequence {Pan }n can be taken to be in (N, A). Now as (ϕ, e)|(N,A) is soft continuous at each Paxnn and e there exists, for each n, a soft point P tn ∈ e (N, A) is soft dense in X, e (M, A) ∩ bn e xn , P t n ) < e (N, A) such that d(P e n1̄ . e n1̄ and ρe((ϕ, e)(Paxnn ), (ϕ, e)(Pbtnn )) < (M, A) ∩ an bn e xn , P t n ) < e n1̄ implies Pbtnn −→ Pax . Again since Therefore, Paxnn −→ Pax and d(P an bn (ϕ, e)|(M,A) is soft continuous at Pax , (ϕ, e)(Pbtnn ) −→ (ϕ, e)(Pax ) and therefore by, e n1̄ we get (ϕ, e)(Paxnn ) −→ (ϕ, e)(Pax ), which contraρe((ϕ, e)(Paxnn ), (ϕ, e)(Pbtnn )) < e e dicts our assumption ρe((ϕ, e)(Paxnn ), (ϕ, e)(Pax )) ≥ ǫ for every n. Hence (ϕ, e) is x soft continuous at Pa .  Next, we give the following characterization of soft continuity of a soft map in terms of convergent sequence of soft points taken from a soft dense subset of domain of the soft map: e A) and (Ye , ρe, B) be two soft metric spaces. For a soft e d, Theorem 3.5. Let (X, e A) → (Ye , ρe, B), the following conditions are equivalent. e d, mapping, (ϕ, e) : (X, (1) (ϕ, e) is soft continuous. e deZ , A) → (Ye , ρe, B) is soft continuous and (ϕ, e) is soft (2) (ϕ, e)|(Z,A) : (Z, continuous at each point of (Z, A)c . (3) for any sequence of soft points {Paxnn }n in (Z, A) converging to a soft point e implies {(ϕ, e)(P xn )}n converges to (ϕ, e)(P x ). eX Pax ∈ an a Proof. (1) ⇔ (2) by Theorem 3.2 (2) ⇒ (3) is obvious. 3628 S. KAUR, A. SINGH, AND N. S. NOORIE e such that P xn (3) ⇒ (1) Suppose {Paxnn }n be a sequence of soft points in X an e Since (Z, A) is soft dense in X, e for each n, there e X. −→ Pax where Pax ∈ dn e Theree X. is sequence {Pankk } of soft points in (Z, A) converging to Paxnn ∈ dn fore by (3), (ϕ, e)(Pankk ) −→ (ϕ, e)(Paxnn ). Then for each n, there exist a posie adnnk , P xn ) < e n1̄ for all k ≥ k1 (n) and tive integers k1 (n) and k2 (n) such that d(P an k dn e n1̄ for all k ≥ k2 (n). Let k(n) = max{k1 (n), k2 (n)}, ρe((ϕ, e)(Pankk ), (ϕ, e)(Paxnn )) < e adnnk(n) , P xn ) −→ 0̄ and ρe((ϕ, e)(Padnnk(n) ), (ϕ, e)(P xn )) −→ 0̄ as n −→ ∞. Now d(P k(n) as Paxnn an −→ Pax , k(n) we get dnk(n) Pank(n) −→ Pax . an dn k(n) Therefore, by (3) again, (ϕ, e)(Pank(n) ) dn k(n) ), (ϕ, e)(Paxnn )) −→ 0̄ implies that (ϕ, e)(Paxnn ) −→ (ϕ, e)(Pax ) and so ρe((ϕ, e)(Pank(n) −→ (ϕ, e)(Pax ). Hence (ϕ, e) is soft continuous.  By using above theorem we get the following corollary. e dez , A) → (Ye , ρe, B) be a soft continuous mapping Corollary 3.1. Let (ϕ, e) : (Z, e Then there exist a largest soft subset (W, A) of X, e where (Z,A) is soft dense in X. f , dew , A) → e (Z, A)⊆(W, A) such that (ϕ, e) can be extended to a soft map (f, e) : (W (Ye , ρe, B) which is soft continuous. ′ e Pax and (W, A) = ∪ e Pax where S be set of all soft Proof. Let (Z , A) = (Z, A)∪ x Pa ∈S e for which there exist a soft continuous extension of (ϕ, e), (fx , e) : points in X f′ , de ′ , A) → (Ye , ρe, B). Obviously, (Z, A)⊆(W, f , dew , A) → e A). Let (f, e) : (W (Z z (Ye , ρe, B) defined by (f, e)(Pax ) = (fx , e)(Pax ) is the extension of (ϕ, e) to (W,A) e (Z, A). As and {Padnn }n be a sequence of soft points converging to Pax where Padnn ∈ dn dn x (fx , e) is soft continuous, (f, e)(Pan ) = (fx , e)(Pan ) −→ (fx , e)(Pa ) = (f, e)(Pax ). e Now, Let By Theorem 3.5 (3), (f, e) is soft continuous as (Z,A) is soft dense in X. f0 , dew0 , A) → e 0 , A) and (f0 , e) : (W (W0 , A) be another soft set such that (Z, A)⊆(W e de ′ , A) → (Ye , ρe, B) be a soft continuous extension of (ϕ, e) then (ϕ, e)|(Z ′ ,A) : (Z, Z xe e e (Y , ρe, B) is soft continuous for all Pa ∈(W0 , A) and so (W0 , A)⊆(W, A) then (W, A) e where (Z, A)⊆(W, e is largest soft subset of X A).  Pax In the following corollary of above proved Theorem 3.2, we obtain the soft continuity of a soft map in terms of the soft continuity of its restrictions to members of a soft cover of the domain having soft dense intersection: APPLICATIONS OF SOFT DENSE SETS TO SOFT CONTINUITY 3629 e that is X e = ∪ e (F, A)α Corollary 3.2. Let {(F, A)α | α ∈ Λ} be soft cover of X, α∈Λ e A) → (Ye , ρe, B) is e then a map (ϕ, e) : (X, e d, e (F, A)α is soft dense in X such that ∩ α soft continuous if (ϕ, e)|(F,A)α is soft continuous. Proof. By Theorem 3.5, it is sufficient to prove that if the sequence {Padnn }n of e then (ϕ, e)(P dn ) −→ (ϕ, e)(P x ). As e (F, A)α converging to Pax ∈ eX soft points in ∩ an a α e Px ∈ e (F, A)α for some α, result follows {(F, A)α | α ∈ Λ} be soft cover of X, a x from soft continuity of (ϕ, e)|(F,A)α at Pa .  4. C ONCLUSION In this paper, we give applications of soft dense sets to soft continuous mappings between soft metric spaces. 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