Intuitionistic fuzzy-γ-retracts and interval-valued intuitionistic almost (near) compactness

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1 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, Available online at Intuitionisti fuzzy-γ-retrats and interval-valued intuitionisti almost (near) ompatness Mohammed M. Khalaf a, Sayer Obaid Alharbi b, and Wathek Chammam b,* a Department of Mathematis, Higher Institute of Engineering and Tehnology King Marriott, Alexandria, P.O. Box 3135, Egypt b Department of Mathematis, College of Siene Al-Zulfi, Al-Majmaah University, P.O. Box 66, Al-Majmaah 11952, Saudi Arabia Department of Mathematis, Faulty of Sienes of Gabès, Gabès University, Gabès Tunisia Reeived 9 Marh 2018, aepted 30 August 2018, available online 16 November Authors. This is an Open Aess artile distributed under the terms and onditions of the Creative Commons Attribution- NonCommerial 4.0 International Liense ( Abstrat. The aim of this paper is to introdue the onepts of an intuitionisti fuzzy-γ-retrat and an intuitionisti fuzzy-rretrat. Some haraterizations of these new onepts are presented. Examples are given, and properties are established. Also, we study the onepts of interval-valued intuitionisti almost (near) ompatness and define S 1 -regular spaes. We prove that if an intuitionisti fuzzy topologial spae is an S 1 -regular spae and interval-valued intuitionisti almost (near) ompat, then it is interval-valued intuitionisti ompat. Key words: intuitionisti fuzzy-γ-retrat, S 1 -regular spae, interval-valued intuitionisti almost (near) ompatness. 1. INTRODUCTION AND PRELIMINARIES The onept of fuzzy sets was first proposed by Zadeh in 1965 [8]. This onept has a wide range of appliations in various fields suh as omputer engineering, artifiial intelligene, ontrol engineering, operation researh, management siene, robotis, and many more. It gives us a tool to model the unertainty present in a phenomenon that does not have sharp boundaries. Many papers on fuzzy sets have been published, showing their importane and appliations to set theory, algebra, real analysis, measure theory, topology, et. Atanassov [1] extends the fuzzy set haraterized by a membership funtion to the intuitionisti fuzzy set (IFS), whih is haraterized by a membership funtion, a non-membership funtion, and a hesitany funtion. As a result, the IFS an desribe the fuzzy haraters of things in more detail and more omprehensively, whih is found to be more effetive in dealing with vagueness and unertainty. Over the last few deades, the IFS theory has been reeiving more and more attention from researhers and pratitioners, and has been applied to various fields, inluding deision making, logi programming, medial diagnosis, pattern reognition, roboti systems, fuzzy topology, mahine learning, and market predition. Intuitionisti fuzzy sets as a generalization of fuzzy sets an be useful in situations when the desription of a problem by a (fuzzy) linguisti variable, given in terms of a membership funtion only, seems too rough. For example, in deision-making problems, partiularly in the ase of medial diagnosis, sales * Corresponding author, w.hammam@mu.edu.sa

2 388 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, analysis, new produt marketing, et., there is a fair hane of the existene of a non-null hesitation part at eah moment of evaluation of an unknown objet. The onept of an intuitionisti fuzzy set, originally proposed by Atanassov [1], is an important tool for dealing with imperfet and impreise information. Compared with Zadeh s fuzzy sets, an intuitionisti fuzzy set gives the membership and non-membership degree to whih an element belongs to a set. Hene, oping with imperfet and impreise information is more flexible and effetive for intuitionisti fuzzy sets. In reent years, intuitionisti fuzzy set theory has been suessfully applied in many pratial fields, suh as deision analysis and pattern reognition. Combining intuitionisti fuzzy set theory and rough set theory may be a promising topi that deserves further investigation. Some researh has already been arried out on this topi. In 1965, Zadeh presented the idea of a fuzzy sеt [8] as a means to represent unertainty. This notion was originally introdued as a method to onsider impreision and ambiguity ourring in human disourse and thought. Many works by the same author and his olleagues appeared in the literature [3,4]. Later, topologial strutures in fuzzy topologial spaes [5] were generalized to intuitionisti fuzzy topologial spaes by Coker in [4], who then introdued the onept of an intuitionisti set [4]. This onept is the disrete form of an intuitionisti fuzzy set, and it is one of several ways of introduing vagueness in mathematial objets. On the other hand, the onept of a fuzzy retrat was introdued by Rodabaugh [7]. The purpose of this paper is to onstrut the idea of intuitionisti fuzzy retrats, alled IF-R-retrats, whih use the gеnеralizatiοn of intuitionisti fuzzy ontinuity. After giving the fundamental examples, we introdue the onepts of interval-valued intuitionisti almost (near) ompatness and S 1 -regular spaes and prove that if an intuitionisti fuzzy topologial spae ( X, ) is an S 1 -regular spae and interval-valued intuitionisti almost (near) ompat, then it is an interval-valued intuitionisti ompat. Throughout this paper, X denotes a non-empty set. A fuzzy set in X is a funtion with domain X and values in I. The words intuitionisti fuzzy set and intuitionisti fuzzy topologial spae will be abbreviated as IF-set and IF-ts, respetively. Also, by I(ν), C(ν), and ν we will denote respetively the interior, losure, and the omplement of an IF-set ν. A mapping r :(X,δ) ( Y, ) is IF-ontinuous if, r ( ). Let ( X, ) be an IF-ts and A X. Then a maximal subspae (A,δ A ) of ( X, ) is an IF-ts and is defined by A { A : }. Definition 1.1 [1]. Let X be a nonempty set. An IF-set A is an objet of the form A {, x A(), x A(): x x X}, where the funtions µ A : X [0,1] and ν A : X [0,1] denote, respetively, the degree of membership funtion (namely µ A (x)) and the degree of non-membership funtion (namely ν A (x)) of A, 0 A( x) va( x ) 1, for eah x X. Remark 1.1 [2]. Atanassova and Doukovska introdued the following interesting geometrial interpretations to express an IF-set (see Fig. 1). Definition 1.2 [1]. Let X be a nonempty fixed set, and let I be the losed unit interval [0,1]. Consider two IF-sets A{, x A(), x A(): x x X} and B {, x B(), x B(): x x X}. Then (i) A { x, A( x), A( x) : x X}, (ii) A B( A( x) B( x) and A( x) B( x)), for eah x X, (iii) A B A B and B A, (iv) A { x, A( x), A( x)}, and (v) A { x, A( x), A( x)}. Definition 1.3 [6]. Let A be an IF-set of an IF-ts ( X, ). Then A is alled (i) an IF-regular open set if A=I(C(A)), (ii) an IF-semi-open set if A C(I(A)), B A B and B A, (iii) an IF-preopen set if A I(C(A)), (iv) an IF-strongly semi-open set if A I(C(I(A))), and (v) an IF-semi-preopen set if A C(I(C(A))). Their omplements are alled IF-regular losed, IF-semi-losed, IF-prelosed, IF-strongly semi-losed, and IF-semi-prelosed sets, respetively.

3 M. M. Khalaf et al.: Instuitionisti fuzzy-γ-retrats 389 Fig. 1. Geometrial interpretations of an intuitionisti fuzzy set. Definition 1.4. Let f: ( X, ) ( Y, ) be a mapping from an IF-ts ( X, ) to another IF-ts ( Y, ). Then f is alled (vi) an IF-semi-ontinuous mapping if for eah we have f (ν) is an IF-semi-open set of X; (vii) an IF-preontinuous mapping if for eah we have f (ν) is an IF-preopen set of X; (viii) an IF-strongly semi-ontinuous mapping if for eah we have f (ν) is an IF-strongly semiopen set of X; (ix) an IF-semi-preontinuous mapping if for eah we have f (ν) is an IF-semi-preopen set of X. Definition 1.5 [6]. Let ( X, ) be an IF-ts and let A X. Then the IF-subspae ( A, A) is alled an IFretrat (IFR, for short) of ( X, ) if there exists an IF-ontinuous mapping r : ( X, ) ( A, A) suh that r(a)=a for all a A. In this ase r is alled an IF-retration. Definition 1.6 [6]. Let ( X, ) be an IF-ts. Then ( A, A) is said to be an IF-neighbourhood retrat (IFnbd R, for short) of ( X, ) if ( A, A) is an IF-retrat of ( Y, y ) suh that AY X,1 Y. Definition 1.7 [6]. Let ( X, ) be an IF-ts and A X. Then the IF-subspae ( A, A) is alled an IF-semiretrat (IFSR, for short) (respetively, IF-preretrat, IF-strongly semi-retrat, and IF-semi-preretrat) (resp., IFPreR, IFSSR, IFSPR, for short) of ( X, ) if there exists an IF-semiontinuous (resp., IF-preontinuous, IF-strongly semi-ontinuous, IF-semi-preontinuous) mapping r : ( X, ) ( A, A) suh that r(a)=a for all a A. In this ase, f is alled an IF-semi-retration (resp., IF-preretration, IF-strongly semi-retration, IF-semi-preretration). Definition 1.8 [6]. Let ( X, ) be an IF-ts. Then ( A, A) is said to be an IF-neighbourhood semi-retrat (IF-nbd SR, for short) (resp., IF-nbd preretrat, IF-nbd strongly semi-retrat, IF-nbd semi-preretrat) (IF-nbd PreR, IF-nbd SSR, IF-nbd SPR, for short) of ( X, ) if ( A, A) is an IFSR (resp., IFPreR, IFSSR, IFSPR) of (Y,δ Y ) suh that AY X,1 Y. Definition 1.9 [6]. Let f : ( X, ) ( Y, ) be a funtion from an interval-valued intuitionisti fuzzy topologial spae ( It, for short) ( X, ) into an It ( Y, ). Then f is said to be interval-valued intuitionisti-almost open (resp., losed) iff for eah interval-valued intuitionisti fuzzy regular open (resp., losed) set X, f( ) Y. Definition 1.10 [4]. Let ( X, ) be an It. (i) A family { j : j J} of interval-valued intuitionisti fuzzy sets ( Is, for short) of X is alled an interval-valued intuitionisti fuzzy open over ( Io, for short) of X iff jj j 1. (ii) A finite subfamily of an Io G of X whih is also an Io of X is alled a finite subover of G.

4 390 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, (iii) A family M { j : j J} of Is of X satisfies the finite intersetion property (FIP, for short) iff every finite subfamily {λ j 1,λ j 2,λ j 3,,λ jn } of M satisfies the ondition j(1,... n) j 0. (iv) An It ( X, ) is alled interval-valued intuitionisti fuzzy ompat ( I, for short) iff every Io has a finite subover. 2. INTUITIONISTIC FUZZY-γ-RETRACTS In this setion the basi onept of an intuitionisti fuzzy-γ-retrat is introdued, and some haraterizations are presented. Examples and properties are established. Also, the relations between these new onepts are explained. Definition 2.1. Let ( X, ) be an IF-ts and A X. Then a maximal subspae ( A, A) of ( X, ) is alled an IF-γ-retrat of ( X, ) (IF-γ-R, for short) if there exists an IF-γ-ontinuous mapping f : ( X, ) ( A, A) suh that f(x) = x for all x A. In this ase f is alled an IF-γ-retration. Remark 2.1. From the above definitions one may notie that IFR IFSSR IFSR and IFPreR IF-γ-R IFSPR. Example 2.1. Let λ 1 and λ 2 be IF-sets on X ={a, b, }defined by a b a b 1 x,,,,,,, a b a b 2 x,,,,,,, ,1, 1, 2, and A { x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-strongly semiretrat of ( X, ) but not an IF-retrat. Example 2.2. Let λ be an IF-set on X ={a, b, } defined by a b a b x,,,,, ,1,, and A{ x} X. Then ( A, A) is an IF-pre-retrat of ( X, ) but not an IF-strongly semiretrat. Example 2.3. Let λ be an IF-set on X ={a, b, }defined by a b a b x,,,,,,, ,1,, and A{ x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-semi-retrat of ( X, ) but not an IF-strongly semi-retrat. Example 2.4. Let λ be an IF-set on X ={a, b, } defined by a b a b x,,,,,,, ,1,, and A{ x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-semi-preretrat of ( X, ) but not an IF-γ-retrat.

5 M. M. Khalaf et al.: Instuitionisti fuzzy-γ-retrats 391 Example 2.5. Let λ be an IF-set on X={a, b, } defined by a b a b x,,,,,,, ,1,, and A{ x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-γ-retrat of ( X, ) but not an IF-semi-retrat. Example 2.6. Let λ be an IF-set on X ={a, b, } defined by a b a b x,,,,,,, ,1,, and A{ x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-γ-retrat of ( X, ) but not an IF-preretrat. Remark 2.2. Let ( X, ) be an IF-ts and Z Y X. If ( Y, y ) is an IF-γ-retrat of ( X, ) and ( Z,( y ) z) is an IF-γ-retrat of ( Y, y ), then ( Z,( y ) z) need not be an IF-γ-retrat of ( X, ). Example 2.7. Let X { ab,, }, Y{ ab, }, Z { a}, and let λ 1, λ 2 be IF-sets on X defined by a b a b 1 x,,,,,,, a b a b 2 x,,,,,,, ,1, 1, 2. Then ( Y, y ) is an IF-γ-retrat of ( X, ), and ( Z,( y ) z) is an IF-γ-retrat of ( Y, y ), but ( Z,( y ) z) is not an IF-γ-retrat of ( X, ). Y be IF-ts s. If f 1 : X 1 Y 1 is IF-γ-ontinuous and f 2 : X 2 Y 2 is IF-γ-ontinuous, then the produt f 1 f 2 : X 1 X 2 Y 1 Y 2 need not be IF-γ-ontinuous. Remark 2.3. Let ( X1, 1), X2, 2, Y1, 1, and 2, 2 Example 2.8. Let X 1 = Y 1 = { a, b }, 1 0,1, 1, 2, and let ν 1, υ 2 be IF-sets on Y 1, defined by a b a b 1 x,,,,, a b a b 2 x,,,,, a b a b 1 x,,,,, a b a b 2 x,,,,, ,1,,. Let λ 1, λ 2 be IF-sets on X 1 and and let f1 id X 1 : X1 Y1 be defined by f1( x) X, x X1. Then f 1 is IF-γ-ontinuous. Also, let X 2 = Y 2 = {x,y}, 2 {0,1, }, and 2 {0,1, }, Let λ be an IF-set on X 2 and let ν 1 be an IF-set on Y 2, defined by

6 392 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, a b a b x,,,,, a b a b x,,,,, and let f2 id X 2 : X 2 Y2 be defined by f2( x) X, x X2. Then f 2 is IF-γ-ontinuous, but f 1 f 2 need not be IF-γ-ontinuous. Remark 2.4. Let ( X, ) and ( Y, y ) be IF-ts s and A X, B Y. If ( A, A) is an IF-γ-retrat of ( X, ) and ( B, y B ), is an IF-γ-retrat of ( Y, ), then ( A B,( y) AB) need not to be an IF-γ-retrat of ( X Y, ). Remark 2.5. IF-semi-retrats and IF-preretrats are independent onepts. Example 2.9. Let λ 1 and λ 2 be IF-sets on X = {a,b,} defined by a b a b 1 x,,,,,,, a b a b 2 x,,,,,,, ,1, 1, 2, and A{ x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-semi-retrat of ( X, ) but not an IF-preretrat. Example Let λ be an IF-set on X={a,b,} defined by a b a b x,,,,,,, ,1,, and A{ x, y} X, f( a) x, f( b) f( ) y. Then ( A, A) is an IF-preretrat of ( X, ) but not an IF-semi-retrat. 3. IF-R-CONTINUITY AND IF-R-RETRACTS In this setion the basi onepts of intuitionisti fuzzy perfetly retrats, intuitionisti fuzzy R retrats, and intuitionisti fuzzy ompletely retrats and some haraterizations are disussed. Many examples are given, and some properties are established. Also, we define the relations between these new onepts. Definition 3.1. Let f : ( X, ) ( Y, ) be a mapping from an IF-ts ( X, ) to another IF-ts ( Y, ). Then f is alled (i) an IF-perfetly ontinuous (IFPC, for short) mapping if for eah we have f (ν) is both an IFopen and an IF-losed set of X, (ii) an IF-ompletely ontinuous (IFCC, for short) mapping if for eah we have f (ν) is an IFregular open set of X, (iii) an IF-R-ontinuous (IFRC, for short) mapping if for eah IF-regular open we have f (ν) is an IF-regular open set of X. Remark 3.1. The impliations between these different onepts are given by the following diagram: IFPC IFCC IFRC. The onverses of the above impliations need not be true in general, as shown by the following examples.

7 M. M. Khalaf et al.: Instuitionisti fuzzy-γ-retrats 393 Example 3.1. Let X {a,b}, Y {1,2}, Let ( X, ) and ( Y, ) be two IF-ts s where δ {0,1,α,β}, and γ {0,1,θ 1,θ 2 },α,β,θ 1 and θ 2 are defined by f(a) 2, f(b) 1. Then f is IFRC but not IFCC. a b a b x,,,,, a b a b x,,,,, x,,,,, x,,,,, Example 3.2. Let X Y [0,1]. Let ( X, ) and ( Y, ) be two IF-ts s where ,1, C0.7,0.2, C, : 0,0 and 0,1, C, : 0,0, f(x) x. Then f : ( X, ) ( Y, ) is IFCC but not IFPC. Definition 3.2. An IF-ts ( X, ) is alled an IF-extremely disonneted spae (IFED-spae, for short) if the losure of every IF-open set of X is an IF-open set. Lemma 3.1. Let ( X, ) be an IFED-spae. Then, if λ is an IF-regular open set of X, it is both an IF-open set and an IF-losed set. Proof. Let λ be an IF-regular open set of X, then λ I(C(λ)) sine every IF-regular open set is IF-open. Then λ is an IF-open set of X and beause ( X, ) is an IFED-spae, C(λ) λ. Then λ is an IF-losed set. Theorem 3.1. Let ( X, ) be an IFED-spae, and let f : ( X, ) ( Y, ) be a mapping. Then the following are equivalent: (i) f is IFPC, (ii) f is IFCC. Proof. It follows from Lemma 3.1. Theorem 3.2. Let f : ( X, ) ( Y, ) be a mapping. Then f is IFPC (resp., IFCC) iff the inverse image of every IF-losed set of Y is both an IF-open set and an IF-losed (resp., IF-regular open) set of X. Proof. Obvious. Theorem 3.3. Let f : ( X, ) ( Y, ) be a mapping, and let g : X X Y be its graph. If g is IFPC (resp., IFCC) so f is IFPC (resp., IFCC). Proof. Let λ be an IF-open set of Y. Then 1 λ is an IF-open set of X Y. Sine g is IF-perfetly ontinuous, g (1 λ) is both an IF-open set and an IF-losed set of X. Then we have g 1 1 f ( ) f ( ). Therefore f (λ) is both an IF-open set and an IF-losed set of X. Hene f is IFPC. The proof for IFCC is by the same fashion. Definition 3.3. Let ( X, ) be an IF-ts, and let A X. Then the IF-subspae ( A, A) is alled an IF-perfetly retrat (IFPR, for short) (resp., IF-ompletely retrat, IFR-retrat) (resp., IFCR, IFRR, for short) of ( X, ) if there exists an IFPC (resp., IFCC, IFRC) mapping r : ( X, ) ( A, A) suh that r(a) a for all a A. In this ase r is alled an IF-perfetly retration (resp., IF-ompletely retration, IF-R-retration).

8 394 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, Remark 3.1. The impliations between these different onepts are given by the following diagram: IFPR IFCR. The onverse of the above impliation need not be true in general, as shown by the following examples. Example 3.3. Let λ 1 and λ 2 be IF-sets on X {a,b} defined by 0,1, 1, 2, and A { a} X. a b a b 1 x,,,,, a b a b 2 x,,,,, Then ( A, A) is an IFCR of ( X, ) but not an IFPR. Theorem 3.4. Let ( X, ) be an IF-ts, A X and let r : ( X, ) ( A, A) be a mapping suh that r(a) a for all a A. If the graph g : ( X, ) (X A,θ) of r is IFPC (resp., IFCC), then f is an IFretration, where θ is the produt topology generated by δ and δ A. Proof. It follows diretly from Theorem 3.3. Definition 3.4. Let ( X, ) be an IF-ts. Then ( A, A) is said to be an IF-neighbourhood-perfetly retrat (resp., IF-neighbourhood ompletely retrat) (resp., IF-nbd PR, IF-nbd CR, for short) of ( X, ) if ( A, A) is an IFPR (resp., IFCR) of ( Y, y ) suh that AY X,1 Y. Remark 3.2. (i) Every IFPR is an IF-nbd PR, but the onverse is not true. (ii) Every IFCR is an IF-nbd CR, but the onverse is not true. Example 3.4. Let X= { ab,, }, A= { a} X, and let λ 1 and λ 2 be IF-sets on X defined by a b a b 1 x,,,,,,, a b a b 2 x,,,,,, Consider {0,1, 1, 2, 12, 1 2}. Then ( A, A) is an IF-nbd CR of ( X, ) but not an IFCR of ( X, ), and it is an IF-nbd PR of ( X, ) but not an IFPR of ( X, ). 4. INTERVAL-VALUED INTUITIONISTIC COMPACTNESS In this setion we introdue the onepts of interval-valued intuitionisti almost (near) ompatness and define S 1 -regular spaes. We prove that if ( X, ) is an S 1 -regular spae and interval-valued intuitionisti almost (near) ompat, then it is interval-valued intuitionisti ompat. Definition 4.1 [4]. Let ( X, ) be an interval-valued intuitionisti fuzzy topologial spae ( It for short). (i) A family { j : j J} of interval-valued intuitionisti fuzzy open sets of X is alled an Io of X iff jj j1. (ii) A finite subfamily of an Io G of X that is also an Io of X is alled a finite subover of G. (iii) A family M { j : j J} of Io of X satisfies the finite intersetion property iff every finite subfamily { j1,..., jn } of M satisfies the ondition i(1,..., n) ji 0. (iv) An It is alled I iff every Io of X has a finite subover.

9 M. M. Khalaf et al.: Instuitionisti fuzzy-γ-retrats 395 Definition 4.2. (i) An It is alled interval-valued intuitionisti fuzzy almost ompat ( I, for short) iff every Io of X has a finite subolletion whose losures over X. (ii) An It is alled interval-valued intuitionisti fuzzy nearly ompat ( I, for short) iff every Io of nearly X has a finite subolletion suh that the interiors of losures of sets in this subolletion over X. Example 4.1. Let X = I and let { i : i 1,2,3,...} be intuitionisti fuzzy sets defined as follows. First we define i x, i, i and x,, by i i 0.8, x 0 1 ( x) nx, 0< x< n 1 1, < x 1, n 0.1, x 0 1 ( x) 1 nx, 0< x< n 1 0, < x 1, n 0.8, x 0 ( x) 1, otherwise, 0.1, x 0 ( x) 0, otherwise. Seond, we define an intuitionisti fuzzy topologial spae as follows: { 0,1, i, }. Sine { i : i 1,2,3,...} are IF-open sets of X and iji 1, then { i : i 1,2,3,...} is an Io of X. As λ is a finite subfamily of an Io, then i implies that λ is a finite subover of X. Then the intuitionisti fuzzy topologial spae ( X, ) is an intuitionisti fuzzy ompat spae. Theorem 4.1. Let I be an interval-valued intuitionisti fuzzy ompat spae, let I nearly be an intervalvalued intuitionisti fuzzy nearly ompat spae, and let I almost be an interval-valued intuitionisti fuzzy almost ompat spae. Then the impliations between these different onepts are given by the following diagram: I I I. nearly Proof. Let ( X, ) be an I spae, and let { i, i } be an Io of X. Then U U 11 ( x) 0 (1 ( x)) 0 ( x) 0 xx. i i i i i i i Gk 1,..., n i. Gk 1,..., n I Gk 1,..., n Gk1,..., n I( Gk1,..., n) CI( Gk1,..., n) 1, Then Gk 1,..., n is a finite subover suh that We have ( ), therefore and hene ( X, ) is I. For the seond impliation, assuming ( X, ) to be I, we obtain a finite nearly nearly subset Gk 1,..., n suh that CI( Gk 1,..., n) 1, sine Gk1,..., n ( Gk1,..., n) CI( Gk1,..., n) C( Gk1,..., n) 1. It is obvious that CG ( k 1,..., n) 1. Hene, ( X, ) is I.

10 396 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, Theorem 4.2. Let ( X, ),( Y, y) be interval-valued intuitionisti fuzzy-regular spaes. Let f : ( X, ) ( Y, ) be a surjetion and interval-valued intuitionisti fuzzy-almost ontinuous. If ( X, ) is I, then so is ( Y, ). Proof. Let i, i be an Io of Y. Then from the interval-valued intuitionisti fuzzy-almost ontinuity of f it follows that { f IC( i), i } is an Io of X. Sine ( X, ) is I, there exists ( i 1,..., n) suh that i 1,..., n C( f IC( i )) 1. Hene But sine IC i C i that f Ci i1,..., n i i1,..., n i f( C( f IC( ))) ( f( C( f IC( ))) f(1) 1. and from the interval-valued intuitionisti fuzzy-almost ontinuity of f, we see must be an interval-valued intuitionisti fuzzy-almost ontinuous ontaining ( f IC( i )) and hene C( f IC( )). Thus for eah, i whih implies 1. fc( f IC( i)) f( f C( i)) C( i) i C i Hene ( Y, y ) is also I. Theorem 4.3. Let ( X, ),( Y, y) be interval-valued intuitionisti fuzzy topologial spaes. Let f : ( X, ) ( Y, y ) be a surjetion and interval-valued intuitionisti fuzzy-weakly ontinuous. If ( X, ) is an I, then ( Y, ) is I. Proof. Let { i, i } be an Io of Y. Sine f is an IVIF-weakly ontinuous mapping, then we have f ( i) I( f C( i)). Hene, { I( f C( i)), i } is an Io of X. Sine ( X, ) is an I, there exists a finite subover of v indexed by ii 1,2,3,..., n suh that i1,..., nc( f IC( i )) 1. Therefore, Now from i1,..., n i i1,..., n i f( I( f C( ))) f( I( f C( ))) f(1) 1. f( I( f C( i))) f( f C( i)) C( i), i, we dedue f( I( f C( i))) Ci, i. Hene i C( i) 1, whih implies ( Y, y ) is I. Theorem 4.4. Let ( X, ),( Y, y) be interval-valued intuitionisti fuzzy topologial spaes. Let f : ( X, ) ( Y, ) be a surjetion and interval-valued intuitionisti fuzzy-strongly ontinuous. If ( X, ) is an I, then ( Y, ) is I. Proof. Let i, i be an Io of Y. Sine f is IVIF-strongly ontinuous and hene a ontinuous mapping, then we have { f ( i ), i } is an Io of X. Sine ( X, ) is I, there exists a finite subfamily λ i ( i = 1,..., n ) suh that i1,..., nc( f ( i )) 1. From the surjetivity and fuzzy strong ontinuity of f we obtain i1,..., n i i1,..., n i i1,..., n i i1,..., ni f( C( f ( ))) f( C( f C( ))) f( f ( )) f(1) 1. Hene ( Y, ) is I. Theorem 4.5. Let ( X, ) be an interval-valued intuitionisti fuzzy topologial spae. Then the following onditions are equivalent: (i) ( X, ) is I. L U L U (ii) For every family { i, i }, where i x,,,, i i i i for all i, of interval-valued intuitionisti fuzzy regular losed sets suh that ii 0, there exists a finite subfamily λ i ( i = 1,..., n ) suh that i ni i 1,..., 0.

11 M. M. Khalaf et al.: Instuitionisti fuzzy-γ-retrats 397 (iii) ic( i) 0 holds for eah family { i, i } of interval-valued intuitionisti fuzzy regular-regular L U L U open sets where i x,,,, i i i i for all i. (iv) Every interval-valued intuitionisti fuzzy-regular open over of X ontains a finite subfamily whose losures over X. L U L U Proof. (i) (ii) Let x i i i i,,,,, i be a family of interval-valued intuitionisti fuzzy regular-regular losed sets in X with i i 0. Then i i 1. Sine ( i) IC ( ( i) ), we have i IC ( ( i) ) 1. Beause ( X, ) is I almost it follows that there exists a finite subfamily λ i=( 1,..., n ) of λ suh that i 1,..., ncici ( ( ( ) )) 1, therefore i1,..., n CICi i1,..., n CIC i i1,..., nici i i1,..., ni i [ ( ( ( ) ))] [ ( ( ( ) ))] ( ( ( ))) ( ) 0. (ii) (iii) Let { i, i } be a family of interval-valued intuitionisti fuzzy regular-regular open sets, and suppose that i C i 0. Sine { Ci, i } is a family of interval-valued intuitionisti fuzzy regular-regular open sets, there exists a finite subfamily { Ci (1,..., n) } suh that i1,..., nici i1,..., ni 0, whih is a ontradition. (iii) (iv) Let { Ci, i } be a family of interval-valued intuitionisti fuzzy regular-regular open sets overing X. Suppose that i(1,..., n) C( i) 1 for eah finite subover of λ. Then {[ C ( i )] : i } is a family of IVIF-regular open sets, sine [ IC ( ([ C( i)] )] [ II ( ([ C( i)] )] Ii [ C( i)]. Hene i CC [ ( i)] 0 i[ IC ( ( i)] 0, whih is in ontradition with i i 1. (iv) (i) Obvious, sine every interval-valued intuitionisti fuzzy regular-regular open over is an interval-valued intuitionisti fuzzy regular-open over. Theorem 4.6. The image of an I nearly spae under a mapping that is both interval-valued intuitionisti fuzzy regular-almost ontinuous and an interval-valued intuitionisti fuzzy regular-almost open surjetion is an I nearly spae. Proof. The proof of this theorem follows a similar pattern as that of Theorem 4.2. Theorem 4.7. The image of an I nearly spae under an interval-valued intuitionisti fuzzy regular-strong ontinuity is an I spae. Proof. The proof of this theorem follows a similar pattern as that of Theorem 4.2. Definition 4.3. An interval-valued intuitionisti fuzzy topologial spae ( X, ) is an interval-valued intuitionisti fuzzy S 1 -regular spae iff for eah interval-valued intuitionisti fuzzy set X an be written as { IVIFS( X ): C( ) }. Theorem 4.8. An I almost and IVIF-S 1 -regular spae ( X, ) is I. Proof. Suppose that ( X, ) is I. Let i i be an Io of ( X, ). Then ( i) i 1. From the IVIF- S 1 -regularity of ( X, ), it follows Then i : ii : i 1. i { iivifs( X ): C( i) i}. By I, there exists a finite subover C( i) j i( j i), i, j C( i) i, hene ( j) ( Cj) 1, j Therefore ( X, ) is I. that is, 1. suh that C j 1. But Theorem 4.9. An I nearly IVIF-S 1 -regular spae ( X, ) is I. Proof. Suppose that ( X, ) is I. Let nearly i i be an Io of ( X, ). Then i 1. i From the IVIF-S 1 -regularity of ( X, ), it follows that i { iivifs( X ) : C( i) i}. Then i : i i : i 1. By I, there exists a finite subover C nearly i j i ji,, i j suh that j(1,..., n) IC j 1. But ICjCjj 1, hene j(1,..., n) IC j 1. Therefore, ( X, ) is I.

12 398 Proeedings of the Estonian Aademy of Sienes, 2018, 67, 4, CONCLUSIONS The paper introdues the onepts of interval-valued intuitionisti almost (near) ompatness and S 1 -regular spaes and proves that if an intuitionisti fuzzy topologial spae ( X, ) is an S 1 -regular spae and interval-valued intuitionisti almost (near) ompat, then it is interval-valued intuitionisti ompat. The following problems are onsidered in detail: (i) Intuitionisti fuzzy retrats, whih are generalizations using intuitionisti fuzzy ontinuity are defined and ompared, and respetive examples are provided. (ii) Strutural properties of interval-valued intuitionisti almost (near) ompatness and S 1 -regular spaes are disussed via intuitionisti fuzzy topology. (iii) Interval-valued intuitionisti almost (near) ompatness is ompared with other important lasses of interval-valued intuitionisti fuzzy sets, whih provides a way to study ompatness in a more generalized form in the future. ACKNOWLEDGEMENTS The authors are grateful to the reviewer s valuable omments that improved the manusript. Sayer Obaid Alharbi thanks Deanship of Sientifi Researh (DSR) for providing exellent researh failities. The publiation osts of this artile were partially overed by the Estonian Aademy of Sienes. REFERENCES 1. Atanassov, K. Intuitionisti fuzzy sets. Fuzzy Sets Syst., 1986, 20, Atanassova, V. and Doukovska, L. Compass-and-straightedge onstrutions in the intuitionisti fuzzy interpretational triangle: two new intuitionisti fuzzy modal operators. Notes on IFS, 2017, 23, Atanassova, L. Intuitionisti fuzzy impliation 189. Notes on IFS, 2017, 23, Coker, D. An introdution to intuitionisti fuzzy topologial spaes. Fuzzy Sets Syst., 1997, 88, Hanafy, I. M. Completely ontinuous funtions in intuitionisti fuzzy topologial spaes. Czeh. Math. J., 2003, 53(4), Hanafy, I. M., Mahmoud, F. S., and Khalaf, M. M. Intuitionisti fuzzy retrats. Int. J. Log. Intell. Syst., 2005, 5(1), Rodabaugh, S. E. Suitability in fuzzy topologial spaes. J. Math. Anal. Appl., 1981, 79, Zadeh, L. A. Fuzzy sets. Inform. Contr., 1965, 8, Intuitsionistlikud hägusad γ-retraktid ja vahemikväärtustega intuitsionistlik peaaegu kompaktsus Mohammed M. Khalaf, Sayer Obaid Alharbi ja Wathek Chammam On tutvustatud intuitsionistliku hägusa γ-retrakti ja intuitsionistliku hägusa R-retrakti mõistet. On uuritud mõningaid nende uute mõistete vahelisi seoseid, toodud nende mõistete kohta näiteid ja leitud nende mõistete mõningaid omadusi. Samuti on uuritud vahemikväärtustega intuitsionistlikku peaaegu kompaktsust ja defineeritud S 1 -regulaarsed ruumid. On tõestatud, et kui intuitsionistlikult hägus topoloogiline ruum on S 1 -regulaarne ruum ja vahemikväärtustega intuitsionistlikult peaaegu kompaktne, siis on see ka vahemikväärtustega intuitsionistlikult kompaktne.