Journal of the Egyptian Mathematical Society (2014) xxx, xxx-xxx

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

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ORIGINAL ARTICLE

A note on soft connectedness

Sabir Hussain

Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51482, Saudi Arabia Received 30 April 2013; revised 13 January 2014; accepted 2 February 2014

KEYWORDS

Soft topology; Soft open(closed); Soft closure(boundary); Soft mappings; Soft connected

Abstract Soft topological spaces based on soft set theory which is a collection of information granules is the mathematical formulation of approximate reasoning about information systems. In this paper, we define and explore the properties and characterizations of soft connected spaces in soft topological spaces. We expect that the findings in this paper can be promoted to the further study on soft topology to carry out general framework for the practical life applications.

2010 MATHEMATICS SUBJECT CLASSIFICATION: 06D72; 54A40; 54D10

© 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

The researchers introduced the concept of soft sets to deal with uncertainty and to solve complicated problems in economics, engineering, medicines, sociology and environment because of unsuccessful use of classical methods. The well known theories which can be considered as a mathematical tools for dealing with uncertainties and imperfect knowledge are: theory of fuzzy sets [1], theory of intuitionists fuzzy sets [2], theory of vague sets, theory of interval mathematics [3], theory of rough sets and theory of probability [4,5]. All these tools require the pre specification of some parameter to start with.

In 1999 Molodtsov [6] initiated the theory of soft sets as a new mathematical tool to deal with uncertainties while modeling the problems with incomplete information. In [7], he applied successfully in directions such as, smoothness of

functions, game theory, operations research, Riemann-integration, Perron integration, probability and theory of measurement. Maji et. al [8,9] gave first practical application of soft sets in decision making problems.

Many researchers have contributed toward the algebraic structures of soft set theory [10-28]. Shabir and Naz [27] initiated the study of soft topological spaces. They defined basic notions of soft topological spaces such as soft open and soft closed sets, soft subspace, soft closure, soft neighborhood of a point, soft Ti-spaces, for i — 1, 2,3,4, soft regular spaces, soft normal spaces and established their several properties. In 2011, S. Hussain and B. Ahmad [29] continued investigating the properties of soft open(closed), soft neighborhood and soft closure. They also defined and discussed the properties of soft interior, soft exterior and soft boundary. Also in 2012, B. Ahmad and S. Hussain [30] explored the structures of soft topology using soft points. A. Kharral and B. Ahmad [31], defined and discussed the several properties of soft images and soft inverse images of soft sets. They also applied these notions to the problem of medical diagnosis in medical systems. In [32], I. Zorlutana et.al defined and discussed soft pu-continuous mappings. In [33], S. Hussain further established the fundamental and important characterizations of soft pu-continuous func-

E-mail addresses: sabiriub@yahoo.com, sh.hussain@qu.edu.sa Peer review under responsibility of Egyptian Mathematical Society.

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tions, soft pu-open functions and soft pu-closed functions via soft interior, soft closure, soft boundary and soft derived set.

2. Preliminaries

First we recall some definitions and results.

Definition 1 [6]. 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 nonempty 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 2 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.

Definition 2 ([9,14]). For two soft sets (F, A) and (G, B) over a common universe X, we say that (F, A) is a soft subset of (G, B), if

(1) A # B and

(2) for all e 2 A,F(e) and G(e) are identical approximations. We write (F, A) # (G,B).

(F, A) is said to be a soft super set of (G, B), if (G, B) is a soft subset of (F, A). We denote it by (F, A)5(G,B).

Definition 3 [9]. Two soft sets (F, A) and (G, B) over a common universe X are said to be soft equal, if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, a).

Definition 4 [6]. The union of two soft sets of (F, A) and (G, B) over the common universe Xis the soft set (H, C), where C = A U B and for all e 2 C,

{F(e), if e 2 A - B

G(e), if e 2 B - A

F(e)U G(e), if e 2 A n B

We write (F, A)U(G, B) = (H, C).

Definition 5 [6]. The intersection (H, C) of two soft sets (F, A) and (G, B) over a common universe X, denoted (F, A)n(G, B), is defined as C = A n B, and H(e) = F(e)nG(e), for all e 2 C.

Definition 6 [27]. The difference (H, E) of two soft sets (F, E) and (G, E) over X, denoted by (F, E)\(G, E), is defined as H(e) = F(e)\ G(e), for all e 2 E.

Definition 7 [27]. Let (F, E) be a soft set over X and Y be a nonempty subset of X. Then the sub soft set of (F, E) over Y denoted by (YF, E), is defined as follows: FY(a) = YnF(a), for all a 2 E. In other words (YF, E) = Fn(F, E).

Definition 8 [27]. The relative complement of a soft set (F, A) is denoted by (F, A)' and is defined by (F, A)' = (F, A), where F : A ! P(U) is a mapping given by F(a) = U\F(a),foralla 2 A.

Definition 9 [27]. Let s be the collection of soft sets over X, then s is said to be a soft topology on X, if

(1) U, XT belong to s.

(2) the union of any number of soft sets in s belongs to s.

(3) the intersection of any two soft sets in s belongs to s.

The triplet (X, s, E) is called a soft topological space over X.

Definition 10 [27]. Let (X, s, E) be a soft space over X. A soft set (F, E) over X is said to be a soft closed set in X, if its relative complement (F, E)' belongs to s.

Definition 11 [29]. Let (X, s, E) be a soft topological space over X, (G, E) be a soft set over X and x 2 X. Then (G, E) is said to be a soft neighborhood of x, if there exists a soft open set (F, E) such that x 2 (F, E)c(G, E).

Definition 12 [27]. Let (X, s, E) be a soft topological space over X and (F, E) be a soft set over X. Then the soft closure of (F, E), denoted by (F, E) is the intersection of all soft closed super sets of (F, E). Clearly (F, E) is the smallest soft closed set over X which contains (F, E).

Definition 13 [32]. A soft set (F, E) over X is said to be an absolute soft set, denoted by X, if for all e 2 E, F(e) = X. Clearly, Xc = UE and UE = X

Here we consider only soft sets (F, E) over a universe X in which all the parameters of set E are same. We denote the family of these soft sets by SS(X)E.

Definition14 [32]. The soft set (F, E) 2 SS(X)E is called a soft point in X, denoted by eF, if for the element e 2 E, F(e)—/ and F(ec) = / for all ec 2 E \{e}.

Definition 15 [32]. The soft point eF is said to be in the soft set (G, E), denoted by eF2(G, E), if for the element e 2 E and F(e) #G(e).

Definition 16 [32]. A soft set (G, E) in a soft topological space (X, s, E) is called a soft neighborhood (briefly: soft nbd) of the soft point eF 2 X, if there exists a soft open set (H, E) such that

eF 2(H, E) # (G, E).

The soft neighborhood system of a soft point eF, denoted by Ns(eF), is the family of all its soft neighborhoods.

Definition 17 [29]. Let (X, s, E) be a soft topological space and let (G, E) be a soft set over X. The soft interior of soft set (F, E) over X denoted by (F, E)° and is defined as the union of all soft open sets contained in (F, E). Thus (F, E)° is the largest soft open set contained in (F, E).

Definition 18 [29]. Let (X, s, E) be a soft topological space over X. Then soft boundary of soft set (F, E) over X is denoted by (F, E) and is defined as

(F,E) = (F,E)n((F, E)'). Obviously (F,E) is a smallest soft closed set over X containing (F, E).

Definition 19 [29]. (X, s, E) be a soft topological space over X and Y be a nonempty subset of X. Then s Y = { ( Yf, E)|(F, E) 2 s} is said to be the soft relative topology on Y and (Y, sY, E) is called a soft subspace of (X, s, E).

We can easily verify that sY is, in fact, a soft topology on Y.

3. Soft connectedness

Theorem 20 [27]. Let (Y, sY, E) be a soft subspace of soft topological space (X, s, E) and (F, E) be a soft set over X, then

(1) (F, E) is soft open in Y if and only if (F, E) — Yn(G, E), for some (G,E) 2 s.

(2) (F, E) is soft closed in Y if and only if (F, E) — 7n(G, E), for some soft closed set (G,E) in X.

Definition 21 [27]. Let (X, s, E) be a soft topological space over X and x, y 2 X such that x—y. If there exist soft open sets (F, E) and (G, E) such that x 2(F, E), y 2(G, E) and (F, e)(~(G, E) — U, then (X, s, E) is called soft T2-space.

Definition 22 [31]. Let SS(X)E and SS(Y)E be families of soft sets. u : X ! Y and p : E ! E be mappings. Then a mapping fpu : SS(X)E ! SS(Y)E defined as :

(1) Let (F,E) be a soft set in SS(X)E. The image of (F,E) under fpu, written as fm(F,E) — (fpu(F),p(E)), is a soft set in SS(Y)E, such that

8 [ u(F(x)), p-1(y)\A fpu(F)(y) — I xep-1(y)ru

^ /, otherwise,

for all y 2 E.

(2) Let (G,E) be a soft set in SS(V)E. Then the inverse image of (G,E) under fpu, written as f—l(G,E) —

{f-l(G),P-l(E')), is a soft set in SS(U)E such that

fKG'Kx) = {

for all x 2 E.

u-1(G(p(x))), p(x) 2 E, /, otherwise,

Definition 23 [32]. Let fpu : SS(X)E ! SS( Y)E be a mapping and u : X ! Y and p : E ! E be mappings. Then fpu is soft onto, if u : X! Y and p : E ! E are onto and fpu is soft one-one, if u : X ! Y and p : E ! E are one-one.

Definition 24 [32]. Let (X, s, E) and (Y, s*, E) be soft topological spaces. Let u : X ! Y and p : E ! E be mappings. Let fpu : SS(X)E ! SS(Y)E' be a function and eF 2 X.

(a) fpu is soft pu-continuous at eF 2 X, if for each (G,E') 2 Ns*(fpu(eF)), there exists a (H,E) 2 Ns(eF) such that fpu(H,E) # (G,E')._

(b) fpu is soft pu-continuous on X7, if fpu is soft pu-continu-ous at each soft point in X.

Theorem 25 [32]. Let (X, s, E) and (Y, s*, E) be soft topological spaces. Let fpu : SS(X)E ! SS(Y)e be a function. Then the following statements are equivalent.

(a) fpu is soft pu-continuous,

(b) For each (G,E) 2 s*, f-u\(G,E)) 2 s,

(c) For (G,E') soft closed in (Y,s*,E'), fpul(G,E') is soft closed in (X, s,E).

Definition 26. Two soft sets (F, E) and (G, E) over a common universe X are soft disjoint, if (F, E)n(G, E) — U. That is,

/ — F(e)n G(e), for all e 2 E.

Definition 27. Let (X, s, E) be a soft topological space over X. Then (X, s, E) is said to be soft connected, if there does not exist a pair (F, E) and (G, E) of nonempty soft disjoint soft open subsets of (X, s, E) such that X — (F, E)[~(G, E), otherwise (X, s, E) is said to be soft disconnected. In this case, the pair (F, E) and (G, E) is called the soft disconnection of X.

Example 1. Let X —{hi, h2, h3g-the houses under consideration, E = {the set of parameter, each parameter is a sentence or word} = {e1 = beautiful, e2 = cheap} and

s — {U, X, (F1, E), (F2, E), (F3, E), (F4, E), (F5, E)g

where (F1, E), (F2, E), (F3, E), (F4, E) and (F5, E) are soft sets over X which gives us a collection of approximate description of an object, defined as follows:

F1(e1) — {h2}, F1(e2) — {h1}, Fi(e{) — {h2, h3}, F2(e2) — {hu h2}, F3(e1) — {h1, h2}, F3 (e2) — X,

F4(e0 — {h1, h2}, F4(e2) — {hu h3},

F5(e{) — {h2}, F5(e2) — {hu h2}.

Then s defines a soft topology on X and hence (X, s, E) is a soft topological space over X. Clearly X is soft connected.

Definition 28. Let (X, s, E) be a soft topological space over X. A soft subset (F, E) of a soft topological space (X, s, E) is soft connected, if it is soft connected as a soft subspace.

Theorem 29. A soft topological space (X, s, E) is soft discon-nected(respt. soft connected) if and only if there exists (respt. does not exist) nonempty soft subset (F, E) of (X, s, E) which is both soft open and soft closed in (X, s, E).

Theorem 30. Let (X, s, E) and (Y, s*, E) be two soft topological spaces and u : X ! Y and p : E ! E be mappings. Also a soft mapping fpu : SS(X)E ! SS(Y)e is soft pu-continuous and soft onto. If (X, s, E) is soft connected, then the soft image of (X, s, E) is also soft connected.

Proof. Let a soft mappingfpu : SS(X)E ! SS(Y)E be soft pu-continuous and soft onto. Contrarily, suppose that (Y,s*,E ) is soft disconnected and pair (G1, E) and (G2, E ) is a soft disconnections of (Y, s*, E). Since fpu : SS(X)E ! SS(Y)E is soft pu-continuous, therefore f-1(G1, E) and f-1(G2, E) are both soft open in (X,s,E). Clearly the pair f-1(GuE) and f-!(G2,E) is a soft disconnection of (X, s, E), a contradiction. Hence (Y, s*, E) is soft connected. This completes the proof. □

Definition 31. Let (X, s, E) be a soft topological space, (F, E) be soft subset of X, and x 2 X. If every soft neighborhood of x soft intersects (F, E) in some point other than x itself, then x is called a soft limit point of (F, E). The set of all soft limit points of (F, E) is denoted by (F, E)d.

In other words, if (X, s, E) is a soft topological space, (F, E) is a soft subset of X, and x 2 X, then x 2 (F, E)d if and only if (G, E)n((F, E)\{x})—U, for all soft open neighborhoods (g, e) of x.

Remark 1. Form the definition, it follows that x is a soft limit point of (F, E) if and only if x 2 ((F, E)\ {x}).

Theorem 32. Let (X, s, E) be a soft topological space and (F, E) be soft subset of X. Then, (F, E)U (F, E)d = (F, E).

Proof. If x 2 (F, E)U(F, E)d, then x 2 (F, E) or x 2 (F, E)d. If x 2 (F, E), then x 2 (F, E). If x 2(F, E)d, then (G, E)n((F, E)\{x}) — U, for all soft open neighborhoods (G, E) of x, and so (G, E)n(F, E) — U, for all soft open neighborhoods (G, E) of x; hence, x 2 (F, E).

Conversely, if x 2 (F, E), then x 2 (F, E) or x R (F, E). If x 2 (F, E), it is trivial that x 2 (F, E)U(F, E)d. If x R (F, E), then (G, E)n((F, E)\{x}) - U, for all soft open neighborhoods (G, E) of x. Therefore, x 2 (F, E)d implies x 2 (F, E)U(F, E)d. So, (F, E)U(F, E)d = (F, E). Hence the proof. □

Theorem 33. Let (X, s, E) be a soft topological space, and (F, E) be soft subset of X. Then (F, E) is soft closed if and only if (F, E)d # (F, E).

Proof. (F, E) is soft closed if and only if (F, E) = (F, E) if and only if (F, E) = (F, E)U(F, E)d if and only if (F, E)d # (F, E). This completes the proof. □

Theorem 34. Let (X, s, E) be a soft topological space, and (F, E), (G, E) are soft subsets of X. Then,

(1) (F,E) # (G,E) ) (F,E)d # (G,E)d.

(2) ((F,E)n(G,E))d # (F,E)dn(G,E)d.

(3) ((F,E)U(G,E))d = (F,E)dU~(G,E)d.

(4) ((F,E)d) # (F,E)d.

(5) (F, E)d = (F, E)d.

Proof.

(1) Let (F,E) # (G,E). Since (F,E)\{x} # (G,E)\{x}, (F, E)\{x} # (G,E)\{x}, and we obtain (F, E)d # (G, E)d.

(2) ((F,E)n(G,E)) # (F,E) and ((F,E)n(G,E)) # (G,E). Then by (1), ((F,E)n(G,E))d # (F,E)d and ((F,E)n(G,E))d # (G,E)d. Therefore ((F,E)n(G,E))d # (F,E)dn(G,E)d.

(3) For all x 2((F,E)u(G,E))d implies

x 2 ((F, E)U (G, E)) \ {x}. Therefore ((F, E)U(G,E))\{x} = ((F,E)U(G,E)) n {x}c

= ((F, E)n{x}c)U ((G, E)n{x}c) = ((F, E)n{x}c)U (G, E)n{x}c) = ((F, E)\{x})U ((G, E)\{x}) (byTheorem 1 [27]).

if and only x 2 (F,E)dU(G, E)d. Hence ((F,E)U(G,E))d = (F, E)dU (G, E)d. _

(4) Suppose that x R (F,E)d. Then x R (F, E)\{x}. This implies that there is a soft open set (G,E) such that x 2 (G,E) and (G,E)n((F,E)\{x}) = U. We prove that

x R ((F,E)d)d. Suppose on the contrary that x 2 ((F,E)d)d. Then x 2 (F,E)d \{x}. Since x 2 (G,E), we have (G,E)n((F,E)d \{x})-U. Therefore there is y - x such that y 2(G,E)n (F,E)d. It follows that y 2((G,E)\{x})n((F,E)\{y}). Hence

((G,E)\ {x})n((F,E)\{y})-U, a contradiction to the fact that (G,E)n((F,E)\{x}) = U. This implies that

x 2 ((F,E)d)d and so ((F,E)d)d # (F,E)d.

(5) This is a consequence of (2), (3), (5) and Theorem 32.

This completes the proof. □

Now we prove the following theorem:

Theorem 35. If (X, s, E) be a soft T2 space and Y be a nonempty subset of X containing finite number of points, then Y is soft closed.

Proof. Let us take Y = {x}. Now we show that Y is soft closed. If y is a point of X different from x, then x and y have disjoint soft neighborhoods (F, E) and (G, E), respectively. Since (F, E) does not soft intersect {y}, point x cannot belong to the soft closure of the set {y}. As a result, the soft closure of the set {x} is {x} itself, so it is soft closed. Since Y is arbitrary, this is true for all subsets of X containing finite number of points. Hence the proof. □

Next we characterized soft connectedness in terms of soft boundary as:

Theorem 36. A soft topological space (X, s, E) is soft connected if and only if every nonempty proper soft subspace has a nonempty soft boundary.

Proof. Contrarily suppose that a nonempty proper soft subspace (F, E) of a soft connected space (X, s, E) has empty soft boundary. Then (F, E) is soft open and

(F, E)n (X\(F, E)) = U. Let x be a soft limit point of (F, E).

Then x 2 (F, E) but x R (F, E)c. In particular, x R (F, E)c and so x 2 (F, E). Thus (F, E) is soft closed and soft open. By Theorem 29, (X, s, E) is soft disconnected. This contradiction proves that (F, E) has a nonempty soft boundary.

Conversely, suppose that X is soft disconnected. Then by Theorem 29, (X, s, E) has a proper soft subset (F, E) which is both soft closed and soft open. Then (F, A) = (F, A), (F, A)c = (F, A)c and (F, A)n(F, A)c = U. So (F, E) has empty soft boundary, a contradiction. Hence (X, s, E) is soft connected. This completes the proof. □

Theorem 37. Let the pair (F, E) and (G, E) of soft sets be a soft disconnection in soft topological space (X, s, E) and (H, E) be a soft connected subspace of (X, s, E). Then (H, E) is contained in (F, E) or (G, E).

Proof. Contrarily suppose that (H, E) is neither contained in (E,E) nor in (G, E). Then (H,E)n(E, E), (H,E)n(G,E) are both nonempty soft open subsets of (H, E) such that ((H, E)n (E, E))n((H, E)n (G, E)) = U and

((h, e)\(e, e))u((h, e)\(g, e)) = (H, E). This gives that pair of ((H, E)n(E, E)) and ((H, E)n (G, E)) is a soft disconnection of (H, E). This contradiction proves the theorem. □

Theorem 38. Let (G, E) be a so/t connected subset of a so/t topological space (X, s, E) and (E, E) be so/t subset o/ X such that (G, E) # (E, E) # (G, E). Then (E, E) is so/t connected.

Proof. It is sufficient to show that (G, E) is soft connected. On the contrary, suppose that (G, E) is soft disconnected. Then there exists a soft disconnection ((H, E), (K, E)) of (G, E). That is, there are ((H, E)n(G, E)), ((K, E)n (G, E)) soft open sets in (G, E) such that ((H, E)n(G, E))n ((K, E)n (G, E)) = ((H, E)n(K, E))n(G, E) = U, and ((H, E)n (G, E))U((K, E) n(G,E) ) = ((H,E)U(K,E)) n(G,E) = (G,E). This gives that pair ((H, E)n(G, E)) , ((K, E)n(G, E) ) is a soft disconnection of (G, E), a contradiction. This proves that (G, E) is soft connected. Hence the proof. □

Corollary 1. f (F, E) is a soft connected soft subspace of a soft topological space (X, s, E) , then (F, E) is soft connected.

In [27], the soft regular space is defined as:

Definition 39. Let (X, s, E be a soft topological space over X, (G, E be a soft closed set in X and x 2 X such that x R (G, E). If there exist soft open sets (F1, E) and (F2, E) such that x 2(F1,E), (G,E) # (F2, E) and (F1,E)n(F2,E) = U, then (X, s, E is called a soft regular space.

Now we prove the following theorem:

Theorem 40. Let (X, s, E) be a soft regular space and (Y, s Y, E) is a soft subspace of (X, s, E) such that sY = {(Yf,E)|(F,E) 2 s} is soft relative topology on Y. Then (Y, sY, E) is soft regular space.

Proof. Let (Y, sY, E) be a soft subspace of soft regular space (X, s, E). Let y 2 Y and (G, E) be soft closed set in Y such that y R (G, E). Now (G, E)nY = (G, E). Clearly, y R (G, E). Thus (G, E) is soft closed in X such that y R (G, E). Since X is soft regular, so there exist soft open sets (Fi, E) and (F2, E) such that y 2 (F1, E), (G, E) # (F2, E) and (F1, E) \(f2, e) = U. Then Yn(F1, E), Yn(F2, E) are soft disjoint soft open sets in Y such that y 2 Yn (F1, E) and (G, E) # Yn (F2, E). This completes the proof. □

4. Conclusion

During the study toward possible applications in classical and non classical logic, the study of soft sets and soft topology is very important. Soft topological spaces based on soft set theory which is a collection of information granules is the mathematical formulation of approximate reasoning about information systems. Here, we defined and explored the properties of soft connected spaces in soft topological spaces and discussed the behavior of soft connected spaces under soft

pu-continuous mappings. We also characterized soft connectedness in terms of soft boundary and discussed the behavior of soft closure of soft connected subspaces. The addition of this concept will also be helpful to strengthen the foundations in the tool box of soft topology. We expect that the findings in this paper can be applied to problems of many fields that contains uncertainties and will promote the further study on soft topology to carry out general framework for the applications in practical life.

Acknowledgment

The author is grateful to the anonymous referees and editor for the detailed and helpful comments that improved this paper.

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