A note on soft connectedness

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

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems

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 Soft Soft Soft Soft

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

ª 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 E-mail addresses: [email protected], [email protected] Peer review under responsibility of Egyptian Mathematical Society.

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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 ¼ 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-

1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2014.02.003 Please cite this article in press as: S. Hussain, A note on soft connectedness, Journal of the Egyptian Mathematical Society (2014), http:// dx.doi.org/10.1016/j.joems.2014.02.003

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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.

e belong to s. (1) U; X (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 ðX; s; EÞ is called a soft topological space over X.

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 PðXÞ denotes the power set of X and A be a nonempty subset of E. A pair ðF; AÞ is called a soft set over X, where F is a mapping given by F : A ! PðXÞ. In other words, a soft set over X is a parameterized family of subsets of the universe X. For e 2 A; FðeÞ may be considered as the set of e-approximate elements of the soft set ðF; AÞ. Clearly, a soft set is not a set. Definition 2 ([9,14]). For two soft sets ðF; AÞ and ðG; BÞ over a common universe X, we say that ðF; AÞ is a soft subset of ðG; BÞ, if (1) A # B and (2) for all e 2 A; F ðeÞ and GðeÞ are identical approxima~ ðG; BÞ. tions. We write ðF ; AÞ # ðF; AÞ is said to be a soft super set of ðG; BÞ, if ðG; BÞ is a ~ soft subset of ðF; AÞ. We denote it by ðF; AÞðG; BÞ. Definition 3 [9]. Two soft sets ðF; AÞ and ðG; BÞ over a common universe X are said to be soft equal, if ðF; AÞ is a soft subset of ðG; BÞ and ðG; BÞ is a soft subset of ðF; AÞ. Definition 4 [6]. The union of two soft sets of ðF; AÞ and ðG; BÞ over the common universe X is the soft set ðH; CÞ, where C ¼ A [ B and for all e 2 C, 8 > < FðeÞ; HðeÞ ¼ GðeÞ; > : FðeÞ [ GðeÞ;

if e 2 A B if e 2 B A if e 2 A \ B

Definition 10 [27]. Let ðX; s; EÞ be a soft space over X. A soft set ðF; EÞ over X is said to be a soft closed set in X, if its relative complement ðF; EÞ0 belongs to s. Definition 11 [29]. Let ðX; s; EÞ be a soft topological space over X; ðG; EÞ be a soft set over X and x 2 X. Then ðG; EÞ is said to be a soft neighborhood of x, if there exists a soft open ~ set ðF; EÞ such that x 2 ðF; EÞðG; EÞ. Definition 12 [27]. Let ðX; s; EÞ be a soft topological space over X and ðF; EÞ be a soft set over X. Then the soft closure of ðF; EÞ, denoted by ðF; EÞ is the intersection of all soft closed super sets of ðF; EÞ. Clearly ðF; EÞ is the smallest soft closed set over X which contains ðF; EÞ. Definition 13 [32]. A soft set ðF; EÞ over X is said to be an e if for all e 2 E; FðeÞ ¼ X. e absolute soft set, denoted by X, c c e e Clearly, X ¼ UE and UE ¼ X. Here we consider only soft sets ðF; EÞ over a universe X in which all the parameters of set E are same. We denote the family of these soft sets by SSðXÞE . Definition 14 [32]. The soft set ðF; EÞ 2 SSðXÞE is called a soft e denoted by eF , if for the element e 2 E, FðeÞ–/ and point in X, c Fðe Þ ¼ / for all ec 2 E n feg. Definition 15 [32]. The soft point eF is said to be in the soft set ~ EÞ, if for the element e 2 E and ðG; EÞ, denoted by eF 2ðG; ~ GðeÞ. FðeÞ #

~ BÞ ¼ ðH; CÞ. We write ðF; AÞ[ðG;

Definition 16 [32]. A soft set ðG; EÞ in a soft topological space ðX; s; EÞ is called a soft neighborhood (briefly: soft nbd) of the soft point eF 2 X, if there exists a soft open set ðH; EÞ such that ~ ðG; EÞ. eF 2 ðH; EÞ #

Definition 5 [6]. The intersection ðH; CÞ of two soft sets ðF; AÞ ~ BÞ, and ðG; BÞ over a common universe X, denoted ðF; AÞ\ðG; ~ is defined as C ¼ A \ B, and HðeÞ ¼ FðeÞ\GðeÞ, for all e 2 C.

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

Definition 6 [27]. The difference ðH; EÞ of two soft sets ðF; EÞ ~ EÞ, is defined as and ðG; EÞ over X, denoted by ðF; EÞnðG; HðeÞ ¼ FðeÞ n GðeÞ, for all e 2 E. Definition 7 [27]. Let ðF; EÞ be a soft set over X and Y be a nonempty subset of X. Then the sub soft set of ðF; EÞ over Y ~ denoted by ðYF ; EÞ, is defined as follows: FY ðaÞ ¼ Y\FðaÞ, e\ðF; ~ EÞ. for all a 2 E. In other words ðYF ; EÞ ¼ Y Definition 8 [27]. The relative complement of a soft set ðF; AÞ is denoted by ðF; AÞ0 and is defined by ðF; AÞ0 ¼ ðF0 ; AÞ, where ~ for all a 2 A. F0 : A ! PðUÞ is a mapping given by F0 ðaÞ ¼ UnFðaÞ, 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

Definition 17 [29]. Let ðX; s; EÞ be a soft topological space and let ðG; EÞ be a soft set over X. The soft interior of soft set ðF; EÞ over X denoted by ðF; EÞ and is defined as the union of all soft open sets contained in ðF; EÞ. Thus ðF; EÞ is the largest soft open set contained in ðF; EÞ. Definition 18 [29]. Let ðX; s; EÞ be a soft topological space over X. Then soft boundary of soft set ðF; EÞ over X is denoted by ðF; EÞ and is defined as ~ ðF; EÞ ¼ ðF; EÞ\ððF; EÞ0 Þ. Obviously ðF; EÞ is a smallest soft closed set over X containing ðF; EÞ. Definition 19 [29]. ðX; s; EÞ be a soft topological space over X and Y be a nonempty subset of X. Then sY ¼ fðYF ; EÞjðF; EÞ 2 sg is said to be the soft relative topology on Y and ðY; sY ; EÞ is called a soft subspace of ðX; s; EÞ.

Please cite this article in press as: S. Hussain, A note on soft connectedness, Journal of the Egyptian Mathematical Society (2014), http:// dx.doi.org/10.1016/j.joems.2014.02.003

Note on soft connectedness We can easily verify that sY is, in fact, a soft topology on Y. Theorem 20 [27]. Let ðY; sY ; EÞ be a soft subspace of soft topological space ðX; s; EÞ and ðF; EÞ be a soft set over X, then ~ EÞ, (1) ðF ; EÞ is soft open in Y if and only if ðF ; EÞ ¼ Ye \ðG; for some ðG; EÞ 2 s. ~ EÞ, (2) ðF ; EÞ is soft closed in Y if and only if ðF ; EÞ ¼ Ye \ðG; for some soft closed set ðG; EÞ in X.

Definition 21 [27]. Let ðX; s; EÞ be a soft topological space over X and x; y 2 X such that x–y. If there exist soft open sets ðF; EÞ and ðG; EÞ such that x 2 ðF; EÞ; y 2 ðG; EÞ and ~ EÞ ¼ U, then ðX; s; EÞ is called soft T2 -space. ðF; EÞ\ðG; Definition 22 [31]. Let SSðXÞE and SSðYÞE0 be families of soft sets. u : X ! Y and p : E ! E0 be mappings. Then a mapping fpu : SSðXÞE ! SSðYÞE0 defined as : (1) Let ðF ; EÞ be a soft set in SSðX ÞE . The image of ðF ; EÞ under fpu , written as fpu ðF ; EÞ ¼ ðfpu ðF Þ; pðEÞÞ, is a soft set in SSðY ÞE0 such that 8 [ uðFðxÞÞ; p1 ðyÞ \ A–/; < fpu ðFÞðyÞ ¼ x2p1 ðyÞ\A : /; otherwise; for all y 2 E0 . (2) Let ðG; E0 Þ be a soft set in SSðV ÞE0 . Then the inverse image of ðG; E0 Þ under fpu , written as fpu1 ðG; E0 Þ ¼ fpu1 ðGÞ; p1 ðE0 Þ , is a soft set in SSðU ÞE such that 1 u ðGðpðxÞÞÞ; pðxÞ 2 E0 ; f1 ðGÞðxÞ ¼ pu /; otherwise; for all x 2 E.

Definition 23 [32]. Let fpu : SSðXÞE ! SSðYÞE0 be a mapping and u : X ! Y and p : E ! E0 be mappings. Then fpu is soft onto, if u : X ! Y and p : E ! E0 are onto and fpu is soft one-one, if u : X ! Y and p : E ! E0 are one-one. Definition 24 [32]. Let ðX; s; EÞ and ðY; s ; E0 Þ be soft topological spaces. Let u : X ! Y and p : E ! E0 be mappings. Let e fpu : SSðXÞE ! SSðYÞE0 be a function and eF 2 X. e (a) fpu is soft pu-continuous at eF 2 X , if for each ðG; E0 Þ 2 N s ðfpu ðeF ÞÞ, there exists a ðH ; EÞ 2 N s ðeF Þ ~ ðG; E0 Þ. such that fpu ðH ; EÞ # e , if fpu is soft pu-continu(b) fpu is soft pu-continuous on X e. ous at each soft point in X Theorem 25 [32]. Let ðX; s; EÞ and ðY; s ; E0 Þ be soft topological spaces. Let fpu : SSðXÞE ! SSðYÞE0 be a function. Then the following statements are equivalent. (a) fpu is soft pu-continuous, (b) For each ðG; E0 Þ 2 s , fpu1 ððG; E0 ÞÞ 2 s, (c) For ðG; E0 Þ soft closed in ðY ; s ; E0 Þ; fpu1 ðG; E0 Þ is soft closed in ðX ; s; EÞ.

3 3. Soft connectedness Definition 26. Two soft sets ðF; EÞ and ðG; EÞ over a common ~ EÞ ¼ U. That is, universe X are soft disjoint, if ðF; EÞ\ðG; / ¼ FðeÞ \ GðeÞ, for all e 2 E. Definition 27. Let ðX; s; EÞ be a soft topological space over X. Then ðX; s; EÞ is said to be soft connected, if there does not exist a pair ðF; EÞ and ðG; EÞ of nonempty soft disjoint soft e ¼ ðF; EÞ[ðG; ~ EÞ, otheropen subsets of ðX; s; EÞ such that X wise ðX; s; EÞ is said to be soft disconnected. In this case, the pair ðF; EÞ and ðG; EÞ is called the soft disconnection of X. Example 1. Let X ¼ fh1 ; h2 ; h3 g-the houses under consideration, E = {the set of parameter, each parameter is a sentence or word} = {e1 = beautiful, e2 = cheap} and e ðF1 ; EÞ; ðF2 ; EÞ; ðF3 ; EÞ; ðF4 ; EÞ; ðF5 ; EÞg s ¼ fU; X; where ðF1 ; EÞ; ðF2 ; EÞ; ðF3 ; EÞ; ðF4 ; EÞ and ðF5 ; EÞ are soft sets over X which gives us a collection of approximate description of an object, defined as follows: F1 ðe1 Þ ¼ fh2 g; F1 ðe2 Þ ¼ fh1 g; F2 ðe1 Þ ¼ fh2 ; h3 g; F2 ðe2 Þ ¼ fh1 ; h2 g; F3 ðe1 Þ ¼ fh1 ; h2 g; F3 ðe2 Þ ¼ X; F4 ðe1 Þ ¼ fh1 ; h2 g; F4 ðe2 Þ ¼ fh1 ; h3 g; F5 ðe1 Þ ¼ fh2 g; F5 ðe2 Þ ¼ fh1 ; h2 g: Then s defines a soft topology on X and hence ðX; s; EÞ is a soft topological space over X. Clearly X is soft connected. Definition 28. Let ðX; s; EÞ be a soft topological space over X. A soft subset ðF; EÞ of a soft topological space ðX; s; EÞ is soft connected, if it is soft connected as a soft subspace. Theorem 29. A soft topological space ðX; s; EÞ is soft disconnected(respt. soft connected) if and only if there exists (respt. does not exist) nonempty soft subset ðF; EÞ of ðX; s; EÞ which is both soft open and soft closed in ðX; s; EÞ. Theorem 30. Let ðX; s; EÞ and ðY; s ; E0 Þ be two soft topological spaces and u : X ! Y and p : E ! E0 be mappings. Also a soft mapping fpu : SSðXÞE ! SSðYÞE0 is soft pu-continuous and soft onto. If ðX; s; EÞ is soft connected, then the soft image of ðX; s; EÞ is also soft connected. Proof. Let a soft mapping fpu : SSðXÞE ! SSðYÞE0 be soft pucontinuous and soft onto. Contrarily, suppose that ðY; s ; E0 Þ is soft disconnected and pair ðG1 ; E0 Þ and ðG2 ; E0 Þ is a soft disconnections of ðY; s ; E0 Þ. Since fpu : SSðXÞE ! SSðYÞE0 is soft pu0 0 1 continuous, therefore f1 pu ðG1 ; E Þ and fpu ðG2 ; E Þ are both soft 0 0 1 open in ðX; s; EÞ. Clearly the pair f1 pu ðG1 ; E Þ and fpu ðG2 ; E Þ is a soft disconnection of ðX; s; EÞ, a contradiction. Hence ðY; s ; E0 Þ is soft connected. This completes the proof. h Definition 31. Let ðX; s; EÞ be a soft topological space, ðF; EÞ be soft subset of X, and x 2 X. If every soft neighborhood of x soft intersects ðF; EÞ in some point other than x itself, then x is called a soft limit point of ðF; EÞ. The set of all soft limit points of ðF; EÞ is denoted by ðF; EÞd .


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In other words, if ðX; s; EÞ is a soft topological space, ðF; EÞ is a soft subset of X, and x 2 X, then x 2 ðF; EÞd if and only if ~ for all soft open neighborhoods ~ ðG; EÞ\ððF; EÞ n fxgÞ–U, ðG; EÞ of x. Remark 1. Form the definition, it follows that x is a soft limit ~ point of ðF; EÞ if and only if x 2 ððF; EÞnfxgÞ. Theorem 32. Let ðX; s; EÞ be a soft topological space and ðF; EÞ ~ EÞd ¼ ðF; EÞ. be soft subset of X. Then, ðF; EÞ[ðF; ~ EÞd , then x 2 ðF; EÞ or x 2 ðF; EÞd . If Proof. If x 2 ðF; EÞ[ðF; x 2 ðF; EÞ, then x 2 ðF; EÞ. If x 2 ðF; EÞd , then ~ ðG; EÞ\ððF; EÞ n fxgÞ – U, for all soft open neighborhoods ~ EÞ – U, for all soft open neighðG; EÞ of x, and so ðG; EÞ\ðF; borhoods ðG; EÞ of x; hence, x 2 ðF; EÞ. Conversely, if x 2 ðF; EÞ, then x 2 ðF; EÞ or x R ðF; EÞ. If ~ EÞd . If x R ðF; EÞ, x 2 ðF; EÞ, it is trivial that x 2 ðF; EÞ[ðF; ~ then ðG; EÞ\ððF; EÞ n fxgÞ – U, for all soft open neighborhoods ðG; EÞ of x. Therefore, x 2 ðF; EÞd implies ~ EÞd . So, ðF; EÞ[ðF; ~ EÞd ¼ ðF; EÞ. Hence the x 2 ðF; EÞ[ðF; proof. h Theorem 33. Let ðX; s; EÞ be a soft topological space, and ðF; EÞ be soft subset of X. Then ðF; EÞ is soft closed if and only if ~ ðF; EÞ. ðF; EÞd # Proof. ðF; EÞ is soft closed if and only if ðF; EÞ ¼ ðF; EÞ if and ~ ðF; EÞ. ~ EÞd if and only if ðF; EÞd # only if ðF; EÞ ¼ ðF; EÞ[ðF; This completes the proof. h Theorem 34. Let ðX; s; EÞ be a soft topological space, and ðF; EÞ; ðG; EÞ are soft subsets of X. Then, ~ ðG; EÞ ) ðF ; EÞd # ~ ðG; EÞd . (1) ðF ; EÞ # d d ~ ðF ; EÞ \ðG; ~ EÞÞ # ~ EÞd . (2) ððF ; EÞ\ðG; d d ~ EÞÞ ¼ ðF ; EÞ [ðG; ~ EÞd . (3) ððF ; EÞ[ðG; d d ~ d (4) ððF ; EÞ Þ # ðF ; EÞ . (5) ðF ; EÞd ¼ ðF ; EÞd .

Proof.

~ EÞd . Hence ððF; EÞ[ðG; ~ EÞÞd ¼ if and only x 2 ðF; EÞd [ðG; d d ~ ðF; EÞ [ðG; EÞ . (4) Suppose that x R ðF ; EÞd . Then x R ðF ; EÞ n fxg. This implies that there is a soft open set ðG; EÞ such that ~ ; EÞ n fxgÞ ¼ U. We prove that x 2 ðG; EÞ and ðG; EÞ\ððF d

x R ððF ; EÞd Þ .

Suppose

d d

on

the

contrary

that

d

x 2 ððF ; EÞ Þ . Then x 2 ðF ; EÞ n fxg. Since x 2 ðG; EÞ, ~ ; EÞd n fxgÞ–U. Therefore there is we have ðG; EÞ\ððF ~ ; EÞd . It follows that y – x such that y 2 ðG; EÞ\ðF ~ ; EÞ n fygÞ. y 2 ððG; EÞ n fxgÞ\ððF Hence ~ ; EÞ n fygÞ–U, a contradiction to the ððG; EÞ n fxgÞ\ððF ~ ; EÞ n fxgÞ ¼ U. This implies that fact that ðG; EÞ\ððF d d d ~ ðF ; EÞd . x 2 ððF ; EÞ Þ and so ððF ; EÞd Þ # (5) This is a consequence of (2), (3), (5) and Theorem 32. This completes the proof.

h

Now we prove the following theorem: Theorem 35. If ðX; s; EÞ 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 ¼ fxg. 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 ðF; EÞ and ðG; EÞ, respectively. Since ðF; EÞ does not soft intersect fyg, point x cannot belong to the soft closure of the set fyg. As a result, the soft closure of the set fxg is fxg 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. h Next we characterized soft connectedness in terms of soft boundary as: Theorem 36. A soft topological space ðX; s; EÞ 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 ðF; EÞ of a soft connected space ðX; s; EÞ has empty soft boundary. Then ðF; EÞ is soft open and ~ n ðF; EÞÞ ¼ U. Let x be a soft limit point of ðF; EÞ. ðF; EÞ\ðX

~ ðG; EÞ. Since ðF ; EÞ n fxg # ~ ðG; EÞ n fxg, (1) Let ðF ; EÞ # ~ ðG; EÞ n fxg, ðF ; EÞ n fxg # and we obtain ~ ðG; EÞd . ðF ; EÞd # ~ ðF ; EÞ and ððF ; EÞ\ðG; ~ ðG; EÞ. ~ EÞÞ # ~ EÞÞ # (2) ððF ; EÞ\ðG; d d ~ ~ EÞÞ # ðF ; EÞ Then by (1), ððF ; EÞ\ðG; and ~ ðG; EÞd . Therefore ððF ; EÞ\ðG; ~ EÞÞd # ~ EÞÞd ððF ; EÞ\ðG; ~ ðF ; EÞd \ðG; ~ EÞd . # ~ EÞÞd (3) For all x 2 ððF ; EÞ[ðG; implies ~ EÞÞ n fxg. Therefore x 2 ððF ; EÞ[ðG;

Then x 2 ðF; EÞ but x R ðF; EÞc . In particular, x R ðF; EÞc and so x 2 ðF; EÞ. Thus ðF; EÞ is soft closed and soft open. By Theorem 29, ðX; s; EÞ is soft disconnected. This contradiction proves that ðF; EÞ has a nonempty soft boundary. Conversely, suppose that X is soft disconnected. Then by Theorem 29, ðX; s; EÞ has a proper soft subset ðF; EÞ which is both soft closed and soft open. Then ðF; AÞ ¼ ðF; AÞ, ~ AÞc ¼ U. So ðF; EÞ has empty ðF; AÞc ¼ ðF; AÞc and ðF; AÞ\ðF; soft boundary, a contradiction. Hence ðX; s; EÞ is soft connected. This completes the proof. h

~ EÞÞ n fxg ¼ ððF; EÞ [ ðG; EÞÞ \ fxgc ððF; EÞ[ðG; ~ c Þ[ððG; ~ ~ cÞ ¼ ððF; EÞ\fxg EÞ\fxg ~ c Þ[ðG; ~ EÞ\fxg ~ cÞ ¼ ððF; EÞ\fxg ~ ¼ ððF; EÞ n fxgÞ[ððG; EÞ n fxgÞ

Theorem 37. Let the pair ðF; EÞ and ðG; EÞ of soft sets be a soft disconnection in soft topological space ðX; s; EÞ and ðH; EÞ be a soft connected subspace of ðX; s; EÞ. Then ðH; EÞ is contained in ðF; EÞ or ðG; EÞ.

ðbyTheorem1½27Þ:


Note on soft connectedness Proof. Contrarily suppose that ðH; EÞ is neither contained in ~ EÞ; ðH; EÞ\ðG; ~ EÞ are ðF; EÞ nor in ðG; EÞ. Then ðH; EÞ\ðF; both nonempty soft open subsets of ðH; EÞ such that ~ EÞÞ\ððH; ~ ~ EÞÞ ¼ U ððH; EÞ\ðF; EÞ\ðG; and ~ EÞÞ[ððH; ~ ~ EÞÞ ¼ ðH; EÞ. This gives that pair ððH; EÞ\ðF; EÞ\ðG; ~ EÞÞ and ððH; EÞ\ðG; ~ EÞÞ is a soft disconnection of ððH; EÞ\ðF; of ðH; EÞ. This contradiction proves the theorem. h Theorem 38. Let ðG; EÞ be a soft connected subset of a soft topological space ðX; s; EÞ and ðF; EÞ be soft subset of X such ~ ðF; EÞ # ~ ðG; EÞ. Then ðF; EÞ is soft connected. that ðG; EÞ # Proof. It is sufficient to show that ðG; EÞ is soft connected. On the contrary, suppose that ðG; EÞ is soft disconnected. Then there exists a soft disconnection ððH; EÞ; ðK; EÞÞ of ðG; EÞ. That ~ EÞÞ, ððK; EÞ\ðG; ~ EÞÞ soft open sets in is, there are ððH; EÞ\ðG; ~ EÞÞ\ððK; ~ ~ EÞÞ ¼ EÞ\ðG; ðG; EÞ such that ððH; EÞ\ðG; ~ EÞÞ\ðG; ~ EÞ ¼ U, and ððH; EÞ\ðG; ~ EÞÞ[ððK; ~ ððH; EÞ\ðK; EÞ ~ EÞÞ\ðG; ~ EÞ ¼ ðG; EÞ. This gives that ~ EÞÞ ¼ ððH; EÞ[ðK; \ðG; ~ EÞÞ, ððK; EÞ\ðG; ~ EÞÞ is a soft disconnection pair ððH; EÞ\ðG; of ðG; EÞ, a contradiction. This proves that ðG; EÞ is soft connected. Hence the proof. h Corollary 1. If ðF; EÞ is a soft connected soft subspace of a soft topological space ðX; s; EÞ, then ðF; EÞ is soft connected. In [27], the soft regular space is defined as: Definition 39. Let ðX; s; EÞ be a soft topological space over X; ðG; EÞ be a soft closed set in X and x 2 X such that x R ðG; EÞ. If there exist soft open sets ðF1 ; EÞ and ðF2 ; EÞ such ~ ðF2 ; EÞ and ðF1 ; EÞ\ðF ~ 2 ; EÞ ¼ U, that x 2 ðF1 ; EÞ; ðG; EÞ # then ðX; s; EÞ is called a soft regular space. Now we prove the following theorem: Theorem 40. Let ðX; s; EÞ be a soft regular space and ðY; sY ; EÞ is a soft subspace of ðX; s; EÞ such that sY ¼ fðYF ; EÞjðF; EÞ 2 sg is soft relative topology on Y. Then ðY; sY ; EÞ is soft regular space. Proof. Let ðY; sY ; EÞ be a soft subspace of soft regular space ðX; s; EÞ. Let y 2 Y and ðG; EÞ be soft closed set in Y such that ~ ¼ ðG; EÞ. Clearly, y R ðG; EÞ. Thus y R ðG; EÞ. Now ðG; EÞ\Y ðG; EÞ is soft closed in X such that y R ðG; EÞ. Since X is soft regular, so there exist soft open sets ðF1 ; EÞ and ðF2 ; EÞ such ~ ðF2 ; EÞ and ðF1 ; EÞ \ ðF2 ; EÞ ¼ U. that y 2 ðF1 ; EÞ; ðG; EÞ # ~ ~ Then Y\ðF1 ; EÞ, Y\ðF2 ; EÞ are soft disjoint soft open sets in ~ Y\ðF ~ 1 ; EÞ and ðG; EÞ # ~ 2 ; EÞ. This comY such that y 2 Y\ðF pletes the proof. h 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

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