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.orgwww.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

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KEYWORDS

Soft topology;

Soft open(closed);

Soft closure(boundary);

Soft mappings;

Soft connected

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Abstract Soft topological spaces based on soft set theory which is a collection of information gran-

ules 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 withuncertainty and to solve complicated problems in economics,

engineering, medicines, sociology and environment becauseof unsuccessful use of classical methods. The well known the-ories which can be considered as a mathematical tools for deal-

ing with uncertainties and imperfect knowledge are: theory offuzzy sets [1], theory of intuitionists fuzzy sets [2], theory of va-gue sets, theory of interval mathematics [3], theory of rough

sets and theory of probability [4,5]. All these tools requirethe pre specification of some parameter to start with.

In 1999 Molodtsov [6] initiated the theory of soft sets as anew mathematical tool to deal with uncertainties while model-

ing the problems with incomplete information. In [7], he ap-plied successfully in directions such as, smoothness of

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tian Mathematical Society.

g by Elsevier

ing by Elsevier B.V. on behalf of E

2.003

ussain, A note on soft connected

functions, game theory, operations research, Riemann-integra-tion, Perron integration, probability and theory of measure-ment. 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] initi-ated the study of soft topological spaces. They defined basic

notions of soft topological spaces such as soft open and softclosed sets, soft subspace, soft closure, soft neighborhood ofa point, soft Ti-spaces, for i ¼ 1; 2; 3; 4, soft regular spaces, softnormal spaces and established their several properties. In 2011,S. Hussain and B. Ahmad [29] continued investigating theproperties of soft open(closed), soft neighborhood and soft

closure. They also defined and discussed the properties of softinterior, soft exterior and soft boundary. Also in 2012, B. Ah-mad and S. Hussain [30] explored the structures of soft topol-

ogy using soft points. A. Kharral and B. Ahmad [31], definedand discussed the several properties of soft images and soft in-verse images of soft sets. They also applied these notions to theproblem of medical diagnosis in medical systems. In [32], I.

Zorlutana et.al defined and discussed soft pu-continuous map-pings. In [33], S. Hussain further established the fundamentaland important characterizations of soft pu-continuous func-

gyptian Mathematical Society.

ness, Journal of the Egyptian Mathematical Society (2014), http://

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.joems.2014.02.003


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2 S. Hussain

tions, soft pu-open functions and soft pu-closed functionsvia soft interior, soft closure, soft boundary and soft derivedset.

2. Preliminaries

First we recall some definitions and results.

Definition 1 [6]. Let X be an initial universe and E be a set ofparameters. 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, asoft set over X is a parameterized family of subsets of the

universe X. For e 2 A;FðeÞ may be considered as the set ofe-approximate elements of the soft set ðF;AÞ. Clearly, a soft setis not a set.

Definition 2 ([9,14]). For two soft sets ðF;AÞ and ðG;BÞ overa common universe X, we say that ðF;AÞ is a soft subset ofðG;BÞ, if

(1) A # B and

(2) for all e 2 A; F ðeÞ and GðeÞ are identical approxima-tions. We write ðF ;AÞ ~# ðG;BÞ.

ðF;AÞ is said to be a soft super set of ðG;BÞ, if ðG;BÞ is asoft subset of ðF;AÞ. We denote it by ðF;AÞ ~�ðG;BÞ.

Definition 3 [9]. Two soft sets ðF;AÞ and ðG;BÞ over a com-mon universe X are said to be soft equal, if ðF;AÞ is a soft sub-

set 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 andðG;BÞ over the common universe X is the soft set ðH;CÞ, whereC ¼ A [ B and for all e 2 C,

HðeÞ ¼FðeÞ; if e 2 A� B

GðeÞ; if e 2 B� A

FðeÞ [ GðeÞ; if e 2 A \ B

8><>:

We write ðF;AÞ ~[ðG;BÞ ¼ ðH;CÞ.

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

Definition 6 [27]. The difference ðH;EÞ of two soft sets ðF;EÞand ðG;EÞ over X, denoted by ðF;EÞ ~nðG;EÞ, is defined asHðeÞ 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 Ydenoted by ðYF;EÞ, is defined as follows: FYðaÞ ¼ Y ~\FðaÞ,for all a 2 E. In other words ðYF;EÞ ¼ eY ~\ðF;EÞ.

Definition 8 [27]. The relative complement of a soft set ðF;AÞ isdenoted by ðF;AÞ0 and is defined by ðF;AÞ0 ¼ ðF0;AÞ, where

F0 : A! PðUÞ isamappinggivenbyF0ðaÞ ¼ U ~nFðaÞ, 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

Please cite this article in press as: S. Hussain, A note on soft connecteddx.doi.org/10.1016/j.joems.2014.02.003

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

Definition 10 [27]. Let ðX; s;EÞ be a soft space over X. A softset ðF;EÞ over X is said to be a soft closed set in X, if its relativecomplement 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 issaid to be a soft neighborhood of x, if there exists a soft openset ðF;EÞ such that x 2 ðF;EÞ ~�ðG;EÞ.

Definition 12 [27]. Let ðX; s;EÞ be a soft topological spaceover 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

absolute soft set, denoted by eX, if for all e 2 E; FðeÞ ¼ eX.

Clearly, eXc ¼ UE and UcE ¼ eX.

Here we consider only soft sets ðF;EÞ over a universe X inwhich all the parameters of set E are same. We denote thefamily of these soft sets by SSðXÞE.

Definition 14 [32]. The soft set ðF;EÞ 2 SSðXÞE is called a soft

point in eX, denoted by eF, if for the element e 2 E, FðeÞ–/ andFðecÞ ¼ / for all ec 2 E n feg.

Definition 15 [32]. The soft point eF is said to be in the soft set

ðG;EÞ, denoted by eF ~2ðG;EÞ, if for the element e 2 E andFðeÞ ~#GðeÞ.

Definition 16 [32]. A soft set ðG;EÞ in a soft topological spaceðX; 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 thateF 2 ðH;EÞ ~# ðG;EÞ.

The soft neighborhood system of a soft point eF, denotedby NsðeFÞ, is the family of all its soft neighborhoods.

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;EÞover X denoted by ðF;EÞ� and is defined as the union of all softopen sets contained in ðF;EÞ. Thus ðF;EÞ� is the largest softopen set contained in ðF;EÞ.

Definition 18 [29]. Let ðX; s;EÞ be a soft topological spaceover X. Then soft boundary of soft set ðF;EÞ over X is denotedby ðF;EÞ and is defined as

ðF;EÞ ¼ ðF;EÞ ~\ððF;EÞ0Þ. Obviously ðF;EÞ is a smallest softclosed set over X containing ðF;EÞ.

Definition 19 [29]. ðX; s;EÞ be a soft topological space over Xand Y be a nonempty subset of X. Then

sY ¼ fðYF;EÞjðF;EÞ 2 sg is said to be the soft relative topologyon Y and ðY; sY;EÞ is called a soft subspace of ðX; s;EÞ.




Note on soft connectedness 3

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 topo-

logical space ðX; s;EÞ and ðF;EÞ be a soft set over X, then

(1) ðF ;EÞ is soft open in Y if and only if ðF ;EÞ ¼ eY ~\ðG;EÞ,for some ðG;EÞ 2 s.

(2) ðF ;EÞ is soft closed in Y if and only if ðF ;EÞ ¼ eY ~\ðG;EÞ,for some soft closed set ðG;EÞ in X.

Definition 21 [27]. Let ðX; s;EÞ be a soft topological spaceover X and x; y 2 X such that x–y. If there exist soft open setsðF;EÞ and ðG;EÞ such that x 2 ðF;EÞ; y 2 ðG;EÞ and

ðF;EÞ ~\ðG;EÞ ¼ U, then ðX; s;EÞ is called soft T2-space.

Definition 22 [31]. Let SSðXÞE and SSðYÞE0 be families of softsets. 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 ;EÞunder fpu, written as fpuðF ;EÞ ¼ ðfpuðF Þ; pðEÞÞ, is a soft

set in SSðY ÞE0 such that

fpuðFÞðyÞ ¼

[x2p�1ðyÞ\A

uðFðxÞÞ; p�1ðyÞ \ A–/;

/; otherwise;

8<:

for all y 2 E0.(2) Let ðG;E0Þ be a soft set in SSðV ÞE0 . Then the inverse

image of ðG;E0Þ under fpu, written as f �1pu ðG;E0Þ ¼f �1pu ðGÞ; p�1ðE0Þ� �

, is a soft set in SSðUÞE such that

f�1pu ðGÞðxÞ ¼u�1ðGðpðxÞÞÞ; pðxÞ 2 E0;

/; otherwise;

�

for all x 2 E.

Definition 23 [32]. Let fpu : SSðXÞE ! SSðYÞE0 be a mappingand u : X! Y and p : E! E0 be mappings. Then fpu is softonto, if u : X! Y and p : E! E0 are onto and fpu is softone-one, if u : X! Y and p : E! E0 are one-one.

Definition 24 [32]. Let ðX; s;EÞ and ðY; s�;E0Þ be soft topolog-ical spaces. Let u : X! Y and p : E! E0 be mappings. Letfpu : SSðXÞE ! SSðYÞE0 be a function and eF 2 eX.

(a) fpu is soft pu-continuous at eF 2 eX , if for each

ðG;E0Þ 2 N s� ðfpuðeF ÞÞ, there exists a ðH ;EÞ 2 N sðeF Þsuch that fpuðH ;EÞ ~# ðG;E0Þ.

(b) fpu is soft pu-continuous on eX , if fpu is soft pu-continu-

ous at each soft point in eX .

� 0
Theorem 25 [32]. Let ðX; s;EÞ and ðY; s ;E Þ be soft topologi-cal spaces. Let fpu : SSðXÞE ! SSðYÞE0 be a function. Then
the following statements are equivalent.

(a) fpu is soft pu-continuous,

(b) For each ðG;E0Þ 2 s�, f �1pu ððG;E0ÞÞ 2 s,(c) For ðG;E0Þ soft closed in ðY ; s�;E0Þ; f �1pu ðG;E0Þ is soft

closed in ðX ; s;EÞ.


3. Soft connectedness

Definition 26. Two soft sets ðF;EÞ and ðG;EÞ over a common

universe X are soft disjoint, if ðF;EÞ ~\ðG;EÞ ¼ U. That is,/ 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 softopen subsets of ðX; s;EÞ such that eX ¼ ðF;EÞ ~[ðG;EÞ, other-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; h3g-the houses under consider-ation, E = {the set of parameter, each parameter is a sentenceor word} = {e1 = beautiful,e2 = cheap} and

s ¼ fU; eX; ðF1;EÞ; ðF2;EÞ; ðF3;EÞ; ðF4;EÞ; ðF5;EÞg

where ðF1;EÞ; ðF2;EÞ; ðF3;EÞ; ðF4;EÞ and ðF5;EÞ are soft

sets over X which gives us a collection of approximate descrip-tion of an object, defined as follows:

F1ðe1Þ ¼ fh2g; F1ðe2Þ ¼ fh1g;F2ðe1Þ ¼ fh2; h3g; F2ðe2Þ ¼ fh1; h2g;F3ðe1Þ ¼ fh1; h2g; F3ðe2Þ ¼ X;

F4ðe1Þ ¼ fh1; h2g; F4ðe2Þ ¼ fh1; h3g;F5ðe1Þ ¼ fh2g; F5ðe2Þ ¼ fh1; h2g:

Then s defines a soft topology on X and hence ðX; s;EÞ is asoft 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 softconnected, if it is soft connected as a soft subspace.

Theorem 29. A soft topological space ðX; s;EÞ is soft discon-nected(respt. soft connected) if and only if there exists (respt.

does not exist) nonempty soft subset ðF;EÞ of ðX; s;EÞ whichis both soft open and soft closed in ðX; s;EÞ.

Theorem 30. Let ðX; s;EÞ and ðY; s�;E0Þ be two soft topological

spaces and u : X! Y and p : E! E0 be mappings. Also a softmapping 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 pu-

continuous and soft onto. Contrarily, suppose that ðY; s�;E0Þ issoft disconnected and pair ðG1;E

0Þ and ðG2;E0Þ is a soft discon-

nections of ðY; s�;E0Þ. Since fpu : SSðXÞE ! SSðYÞE0 is soft pu-continuous, therefore f�1pu ðG1;E

0Þ and f�1pu ðG2;E0Þ are both soft

open in ðX; s;EÞ. Clearly the pair f�1pu ðG1;E0Þ and f�1pu ðG2;E

0Þ isa soft disconnection of ðX; s;EÞ, a contradiction. Hence

ðY; s�;E0Þ is soft connected. This completes the proof. h

Definition 31. Let ðX; s;EÞ be a soft topological space, ðF;EÞbe soft subset of X, and x 2 X. If every soft neighborhoodof 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 limitpoints of ðF;EÞ is denoted by ðF;EÞd.




4 S. Hussain

In other words, if ðX; s;EÞ is a soft topological space, ðF;EÞis a soft subset of X, and x 2 X, then x 2 ðF;EÞd if and only ifðG;EÞ ~\ððF;EÞ n fxgÞ–~U, for all soft open neighborhoodsðG;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;EÞ ~nfxgÞ.

Theorem 32. Let ðX; s;EÞ be a soft topological space and ðF;EÞbe soft subset of X. Then, ðF;EÞ ~[ðF;EÞd ¼ ðF;EÞ.

Proof. If x 2 ðF;EÞ ~[ðF;EÞd, then x 2 ðF;EÞ or x 2 ðF;EÞd. Ifx 2 ðF;EÞ, then x 2 ðF;EÞ. If x 2 ðF;EÞd, thenðG;EÞ ~\ððF;EÞ n fxgÞ – U, for all soft open neighborhoods

ðG;EÞ of x, and so ðG;EÞ ~\ðF;EÞ– U, for all soft open neigh-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. Ifx 2 ðF;EÞ, it is trivial that x 2 ðF;EÞ ~[ðF;EÞd. If x R ðF;EÞ,then ðG;EÞ ~\ððF;EÞ n fxgÞ– U, for all soft open neighbor-

hoods ðG;EÞ of x. Therefore, x 2 ðF;EÞd impliesx 2 ðF;EÞ ~[ðF;EÞd. So, ðF;EÞ ~[ðF;EÞd ¼ ðF;EÞ. Hence theproof. h

Theorem 33. Let ðX; s;EÞ be a soft topological space, and ðF;EÞbe soft subset of X. Then ðF;EÞ is soft closed if and only ifðF;EÞd ~# ðF;EÞ.

Proof. ðF;EÞ is soft closed if and only if ðF;EÞ ¼ ðF;EÞ if andonly if ðF;EÞ ¼ ðF;EÞ ~[ðF;EÞd if and only if ðF;EÞd ~# ðF;EÞ.This completes the proof. h

Theorem 34. Let ðX; s;EÞ be a soft topological space, andðF;EÞ; ðG;EÞ are soft subsets of X. Then,

(1) ðF ;EÞ ~# ðG;EÞ ) ðF ;EÞd ~# ðG;EÞd .(2) ððF ;EÞ ~\ðG;EÞÞd ~# ðF ;EÞd ~\ðG;EÞd .(3) ððF ;EÞ ~[ðG;EÞÞd ¼ ðF ;EÞd ~[ðG;EÞd .(4) ððF ;EÞdÞd ~# ðF ;EÞd .(5) ðF ;EÞd ¼ ðF ;EÞd .

Proof.

(1) Let ðF ;EÞ ~# ðG;EÞ. Since ðF ;EÞ n fxg ~# ðG;EÞ n fxg,ðF ;EÞ n fxg ~# ðG;EÞ n fxg, and we obtain

ðF ;EÞd ~# ðG;EÞd .(2) ððF ;EÞ ~\ðG;EÞÞ ~# ðF ;EÞ and ððF ;EÞ ~\ðG;EÞÞ ~# ðG;EÞ.

Then by (1), ððF ;EÞ ~\ðG;EÞÞd ~# ðF ;EÞd and

ððF ;EÞ ~\ðG;EÞÞd ~# ðG;EÞd . Therefore ððF ;EÞ ~\ðG;EÞÞd

~# ðF ;EÞd ~\ðG;EÞd .(3) For all x 2 ððF ;EÞ ~[ðG;EÞÞd implies

x 2 ððF ;EÞ ~[ðG;EÞÞ n fxg. Therefore

ððF;EÞ ~[ðG;EÞÞ n fxg ¼ ððF;EÞ [ ðG;EÞÞ \ fxgc

¼ ððF;EÞ ~\fxgcÞ ~[ððG;EÞ ~\fxgcÞ¼ ððF;EÞ ~\fxgcÞ ~[ðG;EÞ ~\fxgcÞ¼ ððF;EÞ n fxgÞ ~[ððG;EÞ n fxgÞðbyTheorem1½27�Þ:


if and only x 2 ðF;EÞd ~[ðG;EÞd. Hence ððF;EÞ ~[ðG;EÞÞd ¼ðF;EÞd ~[ðG;EÞd.

(4) Suppose that x R ðF ;EÞd . Then x R ðF ;EÞ n fxg. Thisimplies that there is a soft open set ðG;EÞ such that

x 2 ðG;EÞ and ðG;EÞ ~\ððF ;EÞ n fxgÞ ¼ U. We prove that

x R ððF ;EÞdÞd . Suppose on the contrary that

x 2 ððF ;EÞdÞd . Then x 2 ðF ;EÞd n fxg. Since x 2 ðG;EÞ,we have ðG;EÞ ~\ððF ;EÞd n fxgÞ–U. Therefore there is

y – x such that y 2 ðG;EÞ ~\ðF ;EÞd . It follows that

y 2 ððG;EÞ n fxgÞ ~\ððF ;EÞ n fygÞ. Hence

ððG;EÞ n fxgÞ ~\ððF ;EÞ n fygÞ–U, a contradiction to the

fact that ðG;EÞ ~\ððF ;EÞ n fxgÞ ¼ U. This implies that

x 2 ððF ;EÞdÞd and so ððF ;EÞdÞd ~# ðF ;EÞd .(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 nonemptysubset of X containing finite number of points, then Y is softclosed.

Proof. Let us take Y ¼ fxg. Now we show that Y is softclosed. If y is a point of X different from x, then x and y havedisjoint 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 ofthe 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 connectedif and only if every nonempty proper soft subspace has a non-empty soft boundary.

Proof. Contrarily suppose that a nonempty proper soft

subspace ðF;EÞ of a soft connected space ðX; s;EÞ has emptysoft boundary. Then ðF;EÞ is soft open and

ðF;EÞ ~\ðX n ðF;EÞÞ ¼ U. Let x be a soft limit point of ðF;EÞ.

Then x 2 ðF;EÞ but x R ðF;EÞc. In particular, x R ðF;EÞc andso x 2 ðF;EÞ. Thus ðF;EÞ is soft closed and soft open. By The-

orem 29, ðX; s;EÞ is soft disconnected. This contradictionproves that ðF;EÞ has a nonempty soft boundary.

Conversely, suppose that X is soft disconnected. Then byTheorem 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,ðF;AÞc ¼ ðF;AÞc and ðF;AÞ ~\ðF;AÞc ¼ U. So ðF;EÞ has empty

soft boundary, a contradiction. Hence ðX; s;EÞ is soft con-nected. This completes the proof. h

Theorem 37. Let the pair ðF;EÞ and ðG;EÞ of soft sets be a softdisconnection in soft topological space ðX; s;EÞ and ðH;EÞ be a

soft connected subspace of ðX; s;EÞ. Then ðH;EÞ is contained inðF;EÞ or ðG;EÞ.




Note on soft connectedness 5

Proof. Contrarily suppose that ðH;EÞ is neither contained in

ðF;EÞ nor in ðG;EÞ. Then ðH;EÞ ~\ðF;EÞ; ðH;EÞ ~\ðG;EÞ areboth nonempty soft open subsets of ðH;EÞ such thatððH;EÞ ~\ðF;EÞÞ ~\ððH;EÞ ~\ðG;EÞÞ ¼ U and

ððH;EÞ ~\ðF;EÞÞ ~[ððH;EÞ ~\ðG;EÞÞ ¼ ðH;EÞ. This gives that pairof ððH;EÞ ~\ðF;EÞÞ and ððH;EÞ ~\ðG;EÞÞ is a soft disconnectionof 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 suchthat ðG;EÞ ~# ðF;EÞ ~# ðG;EÞ. Then ðF;EÞ is soft connected.

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. Thatis, there are ððH;EÞ ~\ðG;EÞÞ, ððK;EÞ ~\ðG;EÞÞ soft open sets in

ðG;EÞ such that ððH;EÞ ~\ðG;EÞÞ ~\ððK;EÞ ~\ðG;EÞÞ ¼ððH;EÞ ~\ðK;EÞÞ ~\ðG;EÞ ¼ U, and ððH;EÞ ~\ðG;EÞÞ ~[ððK;EÞ~\ðG;EÞÞ ¼ ððH;EÞ ~[ðK;EÞÞ ~\ðG;EÞ ¼ ðG;EÞ. This gives that

pair ððH;EÞ ~\ðG;EÞÞ, ððK;EÞ ~\ðG;EÞÞ is a soft disconnection

of ðG;EÞ, a contradiction. This proves that ðG;EÞ is soft con-nected. Hence the proof. h

Corollary 1. If ðF;EÞ is a soft connected soft subspace of a softtopological 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 overX; 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 suchthat x 2 ðF1;EÞ; ðG;EÞ ~# ðF2;EÞ and ðF1;EÞ ~\ðF2;EÞ ¼ U,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;EÞis a soft subspace of ðX; s;EÞ such thatsY ¼ fðYF;EÞjðF;EÞ 2 sg is soft relative topology on Y. ThenðY; 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 thaty R ðG;EÞ. Now ðG;EÞ ~\Y ¼ ðG;EÞ. Clearly, y R ðG;EÞ. ThusðG;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 suchthat y 2 ðF1;EÞ; ðG;EÞ ~# ðF2;EÞ and ðF1;EÞ \ ðF2;EÞ ¼ U.Then Y ~\ðF1;EÞ, Y ~\ðF2;EÞ are soft disjoint soft open sets inY such that y 2 Y ~\ðF1;EÞ and ðG;EÞ ~#Y ~\ðF2;EÞ. This com-

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 isvery important. Soft topological spaces based on soft set the-ory which is a collection of information granules is the math-ematical formulation of approximate reasoning about

information systems. Here, we defined and explored the prop-erties of soft connected spaces in soft topological spaces anddiscussed the behavior of soft connected spaces under soft


pu-continuous mappings. We also characterized soft connect-edness in terms of soft boundary and discussed the behaviorof soft closure of soft connected subspaces. The addition of

this concept will also be helpful to strengthen the foundationsin the tool box of soft topology. We expect that the findings inthis paper can be applied to problems of many fields that con-

tains uncertainties and will promote the further study on softtopology to carry out general framework for the applicationsin practical life.

Acknowledgment

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

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A note on soft connectedness