Microsoft Word - 55 542 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 On S*g- -Open Sets In Topological Spaces Sabiha I. Mahmood Jumana S .Tareq Department of Mathematics/College of Science/ University of Al-Mustansiriyah Received in : 9 April 2014 , Accepted in: 1 September 2014 Abstract In this paper, we introduce a new class of sets, namely , s*g-  -open sets and we show that the family of all s*g-  -open subsets of a topological space ),X(  from a topology on X which is finer than  . Also , we study the characterizations and basic properties of s*g-  - open sets and s*g-  -closed sets . Moreover, we use these sets to define and study a new class of functions, namely , s*g-  -continuous functions and s*g-  -irresolute functions in topological spaces . Some properties of these functions have been studied . Keywords: s*g-  -open sets , s*g-  -closed sets , s*g-  -clopen sets , s*g-  -continuous functions , s*g-  -irresolute functions . 543 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Introduction Levine, N. [1,2] introduced and studied semi-open sets and generalized open sets respectively . Njastad, O. [3] , Mashhour, A.S. and et.al. [4] , Andrijevic, D. [5] and Abd El- Monsef,M.E. and et.al [6] introduced α-open sets, pre-open sets ,b-open sets and -open sets respectively . Also, Arya, S.P. and Nour, T.M . [7] , Maki, H. and et.al [8,9], Khan, M. and et.al [10] introduced and investigated generalized semi open sets, generalized α-open sets , α- generalized open sets and s*g-open sets respectively . In this paper, we introduce a new class of sets, namely , s*g-  -open sets and we show that the family of all s*g-  -open subsets of a topological space ),X(  from a topology on X which is finer than  . This class of open sets is placed properly between the class of open sets and each of semi-open sets, α-open sets, pre- open sets, b-open sets, -open sets, generalized semi open sets, generalized α-open sets and α- generalized open sets respectively . Also , we study the characterizations and basic properties of s*g-  -open sets and s*g-  -closed sets . Moreover, we use these sets to define and study a new class of functions, namely , s*g-  -continuous functions and s*g-  -irresolute functions in topological spaces . Some properties of these functions have been studied . Throughout this paper ),X(  , ),Y(  and ),Z(  (or simply X , Y and Z ) represent non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned . 1.Preliminaries First we recall the following definitions and Theorems . Definition(1.1): A subset A of a topological space ),X(  is said to be : i) An semi-open (briefly s-open) set [1] if ))A(int(clA  . ii) An α-open set [3] if )))A(int(clint(A  . iii) An pre-open set [4] if ))A(clint(A  . iv) An b-open set [5] if ))A(int(cl))A(clint(A  . v) An -open set [6] if )))A(cl(int(clA  . The semi-closure (resp. α-closure) of a subset A of a topological space ),X(  is the intersection of all semi-closed (resp. α-closed ) sets which contains A and is denoted by )A(cl s ( resp . )A(cl ) . Clearly )A(cl)A(cl)A(cl s   . Definition(1.2): A subset A of a topological space ),X(  is said to be : i) A generalized closed (briefly g-closed) set [2] if U)A(cl  whenever UA  and U is open in X . ii) A generalized semi-closed (briefly gs-closed) set [7] if U)A(cl s  whenever UA  and U is open in X . iii) A generalized α-closed (briefly gα-closed) set [8] if U)A(cl  whenever UA  and U is α-open in X . iv) An α-generalized closed (briefly αg-closed) set [9] if U)A(cl  whenever UA  and U is open in X . v) An s*g-closed set [10] if U)A(cl  whenever UA  and U is semi-open in X . The complement of a g-closed (resp. gs-closed , gα-closed ,αg-closed , s*g-closed) set is called a g-open (resp. gs-open ,gα-open, αg-open , s*g-open) set . 544 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Definition(1.3): A function ),Y(),X(:f  is called : i) semi-continuous (briefly s-continuous)[1] if )V(f 1 is s-open set in X for every open set Vin Y ii)  -continuous [11] if )V(f 1 is  -open set in X for every open set V in Y . iii) pre-continuous [4] if )V(f 1 is pre-open set in X for every open set V in Y . iv) b-continuous [12] if )V(f 1 is b-open set in X for every open set V in Y . v) -continuous [6] if )V(f 1 is  -open set in X for every open set V in Y . vi) generalized continuous (briefly g-continuous) [13] if )V(f 1 is g-open set in X for every open set V in Y . vii) generalized semi continuous (briefly gs-continuous)[14] if )V(f 1 is gs-open set in X for every open set V in Y . viii) generalized α-continuous (briefly gα-continuous) [8] if )V(f 1 is gα-open set in X for every open set V in Y . ix) α-generalized continuous (briefly αg-continuous) [15] if )V(f 1 is αg-open set in X for every open set V in Y . x) s*g-continuous [16] if )V(f 1 is s*g-open set in X for every open set V in Y . Definition(1.4)[10],[17]: Let ),X(  be a topological space and XA  . Then:- i) The s*g-closure of A , denoted by )A(cl g*s is the intersection of all s*g-closed subsets of X which contains A . ii) The s*g-interior of A , denoted by )A(int g*s is the union of all s*g-open subsets of X which are contained in A . Theorem(1.5)[17]: Let ),X(  be a topological space and XB,A  . Then:- i) )A(cl)A(clA g*s  . ii) A)A(int)Aint( g*s  . iii) If BA  , then )B(cl)A(cl g*sg*s  . iv) A is s*g-closed iff A)A(cl g*s  . v) )A(cl))A(cl(cl g*sg*sg*s  . vi) )AX(cl)A(intX g*sg*s  . vii) )A(clx g*s iff for every s*g-open set U containing x , AU  . viii) )U(cl)U(cl g*sg*s       . Theorem(1.6)[18]: Let YX  be the product space of topological spaces X and Y . If XA  and YB  . Then  )B(cl)A(cl g*sg*s )BA(cl g*s  . 2. Basic Properties Of s*g- -open Sets In this section we introduce a new class of sets, namely , s*g-  -open sets and we show that the family of all s*g-  -open subsets of a topological space ),X(  from a topology on X which is finer than  . 545 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Definition(2.1): A subset A of a topological space ),X(  is called an s*g-  -open set if )))A(int(clint(A g*s . The complement of an s*g-  -open set is defined to be s*g-  -closed . The family of all s*g-  -open subsets of X is denoted by  g*s . Clearly, every open set is an s*g-  -open, but the converse is not true . Consider the following example . Example(2.2): Let }c,b,a{X  and }}a{,,X{  be a topology on X . Then }b,a{ is an s*g-  -open set in X, since })))b,a(int({clint(}b,a{ g*s })a({clint( g*s )Xint( X . But }b,a{ is not open in X . Remark(2.3): s*g-open sets and s*g-  -open sets are in general independent . Consider the following examples:- Example(2.4): Let }c,b,a{X  and },X{  be a topology on X . Then }b{ is an s*g-open set in X , but is not s*g-  -open set , since  })))b(int({clint(}b{ g*s ))(clint( g*s . Also, in example (2.2) }b,a{ is an s*g-  -open set in X , but is not s*g-open , since }c{}b,a{ c  is not s*g-closed set in X , since }c,a{ is an semi-open set in X and }c,a{}c{  , but }c,a{}c,b{})c({cl  . Theorem(2.5): Every s*g-  -open set is  -open (resp. αg-open, gα-open , pre-open, b-open ,  -open ) set . Proof: Let A be any s*g-  -open set in X , then )))A(int(clint(A g*s . Since )))A(int(clint()))A(int(clint( g*s  , thus )))A(int(clint(A  . Therefore A is an  -open set in X . Since every  -open set is αg-open (resp. gα-open , pre-open , b-open , -open ) set .Thus every s*g-  -open set is  -open (resp. αg-open, gα-open , pre-open, b-open , -open ) set . Remark(2.6): The converse of Theorem (2.5) may not be true in general as shown in the following example . Example(2.7): Let }c,b,a{X  & },X{  be a topology on X . Then the set }c,b{ is pre- open (resp. αg-open , gα-open , b-open , -open) in X , but is not s*g-  -open set in X , since })))c,b(int({clint(}c,b{ g*s  )))(clint( g*s . Theorem(2.8): Every s*g-  -open set is semi-open and gs-open set . Proof: Let A be any s*g-  -open set in X , then )))A(int(clint(A g*s . Since ))A(int(cl))A(int(cl)))A(int(clint( g*sg*s  , thus ))A(int(clA  . Therefore A is a semi- open set in X . Since every semi-open set is gs-open set . Thus every s*g-  -open set is semi- open and gs-open set . Remark(2.9): The converse of Theorem (2.8) may not be true in general as shown in the following example . Example(2.10): Let }c,b,a{X  & }}b,a{},b{},a{,,X{  be a topology on X . Then the set }c,a{ is semi-open and gs-open set in X , but is not an s*g-  -open set in X , since  })))c,a(int({clint(}c,a{ g*s })))a({clint( g*s }a{})c,aint({  . Remark(2.11): pre-open sets and αg-open sets are in general independent . Consider the following examples:- Example(2.12): Let ),R(  be the usual topological space . Then the set of all rational numbers Q is a pre-open set , but is not an αg-open set . Also , in Example (2.2) }b{ is an αg-open set , since , }c,a{}b{ c  is an αg-closed set , but is not a pre-open set , since  }))b({clint(}b{ })b,cint({  . 546 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Remark(2.13): g-open sets and gα-open sets are in general independent . Consider the following examples:- Example(2.14): Let }c,b,a{X  & }}c,a{},a{,,X{  be a topology on X . Then the set }b,a{ is a gα-open set in X , since }c{}b,a{ c  is gα-closed , but is not a g-open set in X , since }c{}b,a{ c  is not g-closed . Also, in Example (2.2) }c{ is a g-open set in X , since , }b,a{}c{ c  is g-closed , but is not a gα-open set in X , since }b,a{}c{ c  is not gα-closed . The following diagram shows the relationships between s*g-  -open sets and some other open sets: Proposition(2.15): A subset A of a topological space ),X(  is s*g-  -open if and only if there exists an open subset U of X such that ))U(clint(AU g*s . Proof:  Suppose that A is a s*g-  -open set in X , then )))A(int(clint(A g*s . Since A)Aint(  , thus )))A(int(clint(A)Aint( g*s . Put )Aint(U  , hence there exists an open subset U of X such that ))U(clint(AU g*s . Conversely, suppose that there exists an open subset U of X such that ))U(clint(AU g*s . Since AU   )Aint(U   ))A(int(cl)U(cl g*sg*s   )))A(int(clint())U(clint( g*sg*s  . Since ))U(clint(A g*s , then )))A(int(clint(A g*s . Thus A is an s*g-  -open set in X . Lemma(2.16): Let ),X(  be a topological space . If U is an open set in X , then )AU(cl)A(clU g*sg*s   for any subset A of X . Proof: Let )A(clUx g*s and V be any s*g-open set in X s.t Vx  . Since )A(clx g*s , then by Theorem ((1.5),vii) , AV  . Since VU  is an s*g-open set in X and UVx  , then A)UV(  )AU(V  . Therefore )AU(clx g*s  . Thus )A(clU g*s )AU(cl g*s  for any subset A of X . Theorem(2.17): Let ),X(  be a topological space. Then the family of all s*g-  -open subsets of X from a topology on X . Proof:(i). Since )))(int(clint( g*s  and )))X(int(clint(X g*s , then  g*sX, . (ii). Let  g*sB,A . To prove that  g*sBA  . By Proposition (2.15) , there exists V,U such that ))U(clint(AU g*s and ))V(clint(BV g*s . Notice that VU  and BAVU   . Now ,  ))V(clint())U(clint(BA g*sg*s  ))V(cl))U(clint(int( g*sg*s  547 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 ))V))U(cl(int(clint( g*sg*s  (by Lemma (2.16)) . ))V)U(cl(clint( g*sg*s  ))VU(cl(clint( g*sg*s  (by Lemma (2.16)) . ))VU(clint( g*s  (by Theorem (1.5),v) . Thus ))VU(clint(BAVU g*s   . Therefore by Proposition (2.15) ,  g*sBA  . (iii). Let }:U{  be any family of s*g-  -open subsets of X , then )))U(int(clint(U g*s   for each  . Therefore by Theorem ((1.5) viii) , we get : )))U(int(clint(U g*s      )))U(int(clint( g*s   )))Uint((clint( g*s    )))U(int(clint( g*s    . Hence     g*sU . Thus  g*s is a topology on X . Propositions(2.18): Let ),X(  be a topological space and B be a subset of X . Then the following statements are equivalent: i) B is s*g-  -closed . ii) B)))B(cl((intcl g*s  . iii) There exists a closed subset F of X such that FB))F((intcl g*s  . Proof: )ii()i(  . Since B is an s*g-  -closed set in X  BX  is an s*g-  -open set in X  )))BX(int(clint(BX g*s   )))B(clX(clint(BX g*s  . By Theorem ((1.5), vi) , we get ))B(clX(cl))B(cl(intX g*sg*s  . Hence )))B(cl(intXint(BX g*s  )))B(cl((intclXBX g*s  B)))B(cl((intcl g*s  . )iii()ii(  . Since B)))B(cl((intcl g*s  and )B(clB  , then )B(clB)))B(cl((intcl g*s  . Put )B(clF  , thus there exists a closed subset F of X such that FB))F((intcl g*s  . )i()iii(  . Suppose that there exists a closed subset F of X such that FB))F((intcl g*s  . Hence ))F((intclXBXFX g*s  ))F(intXint( g*s . Since )FX(cl)F(intX g*sg*s  , then ))FX(clint(BXFX g*s  . Hence BX  is an s*g-  -open set in X . Thus B is an s*g-  - closed set in X . Definition(2.19): A subset A of a topological space ),X(  is called an s*g-  -neighborhood of a point x in X if there exists an s*g-  -open set U in X such that AUx  . Remark(2.20): Since every open set is an s*g-  -open set , then every neighborhood of x is an s*g-  -neighborhood of x , but the converse is not true in general . In example (2.2), }b,a{ is an s*g-  -neighborhood of a point b , since }b,a{}b,a{b  . But }b,a{ is not a neighborhood of a point b . Propositions(2.21): A subset A of a topological space ),X(  is s*g-  -open if and only if it is an s*g-  -neighborhood of each of its points . 548 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Proof:  If A is s*g-  -open in X , then AAx  for each Ax  . Thus A is an s*g-  - neighborhood of each of its points . Conversely , suppose that A is an s*g-  -neighborhood of each of its points . Then for each Ax  , there exists an s*g-  -open set xU in X such that AUx x  . Hence AU Ax x    . Since  Ax xUA   , therefore  Ax xUA   . Thus A is an s*g-  -open set in X, since it is a union of s*g-  -open sets . Proposition(2.22): If A is an s*g-  -open set in a topological space ),X(  and )Aint(BA  , then B is an s*g-  -open set in X . Proof: Since A is an s*g-  -open set in X , then by Proposition (2.15), there exists an open subset U of X such that ))U(clint(AU g*s . Since BA   BU  . But ))U(clint()Aint( g*s  ))U(clint(BU g*s . Thus B is an s*g-  -open set in X . Proposition(2.23): If A is an s*g-  -closed set in a topological space ),X(  and AB)A(cl  , then B is an s*g-  -closed set in X . Proof: Since  )A(clXBXAX )AXint(  , then by Proposition (2.22) BX  is an s*g-  -open set in X . Thus B is an s*g-  -closed set in X . Theorem(2.24): A subset A of a topological space ),X(  is clopen (open and closed) if and only if A is s*g-  -clopen (s*g-  -open and s*g-  -closed) . Proof: ( ) . It is a obvious . () . Suppose that A is an s*g-  -clopen set in X , then A is s*g-  -open and s*g-  -closed in X . Hence )))A(int(clint(A g*s and A)))A(cl((intcl g*s  . But by Theorem ((1.5) , i, ii ) we get , )A(cl)A(cl g*s  and )A(int)Aint( g*s , thus : )))A(int(clint(A  and A)))A(cl(int(cl  . Since A)Aint(   )A(cl))A(int(cl  ----------- (1) Since ))A(int(cl)))A(int(clint(  , thus ))A(int(cl)))A(int(clint(A   ))A(int(cl)A(cl  ----------- (2) Therefore from (1) and (2) , we get )A(cl))A(int(cl  ------------ (a) Similarly, since )A(clA   ))A(clint()Aint(  ----------- (3) Now, A)))A(cl(int(cl))A(clint(  , thus )Aint())A(clint(  ------------ (4) Therefore from (3) and (4) , we get )Aint())A(clint(  ----------- (b) Since )Aint())A(clint(   )A(cl))A(int(cl)))A(cl(int(cl  (by (a)) . Since A)))A(cl(int(cl  , then A)A(cl  , but )A(clA  , therefore )A(clA  , hence A is a closed set in X . Similarly, since )A(cl))A(int(cl   )Aint())A(clint()))A(int(clint(  (by (b)) . Since )))A(int(clint(A  , then )Aint(A  , but A)Aint(  , therefore )Aint(A  , hence A is an open set in X . Thus A is a clopen set in X . Definition(2.25): Let ),X(  be a topological space and XA  . Then i) The s*g-  -closure of A , denoted by )A(cl g*s  is the intersection of all s*g-  -closed subsets of X which contains A . 549 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 ii) The s*g-  -interior of A , denoted by )A(int g*s  is the union of all s*g-  -open sets in X which are contained in A . Theorem(2.26): Let ),X(  be a topological space and XB,A  . Then:- i) A)A(int)Aint( g*s   and )A(cl)A(clA g*s   . ii) )A(int g*s  is an s*g-  -open set in X and )A(cl g*s  is an s*g-  -closed set in X . iii) If BA  , then )B(int)A(int g*sg*s   and )B(cl)A(cl g*sg*s   . iv) A is s*g-  -open iff A)A(int g*s  and A is s*g-  -closed iff A)A(cl g*s  . v) )B(int)A(int)BA(int g*sg*sg*s    and )B(cl)A(cl)BA(cl g*sg*sg*s    . vi) )A(int))A((intint g*sg*sg*s   and )A(cl))A(cl(cl g*sg*sg*s   . vii) )A(intx g*s  iff there is an s*g-  -open set U in X s.t AUx  . viii) )A(clx g*s  iff for every s*g-  -open set U containing x , AU  . Proof: It is obvious . Proposition(2.27): Let X and Y be topological spaces . If XA  and YB  . Then BA  is an s*g-  -open set in YX  if and only if A and B are s*g-  -open sets in X and Y respectevely . Proof:  Since A and B are s*g-  -open sets in X and Y respectevely , then by definition (2.1), we get )))A(int(clint(A g*s and )))B(int(clint(B g*s . Hence )))B(int(clint()))A(int(clint(BA g*sg*s  )))B(int(cl))A(int(clint( g*sg*s  . Since  )B(cl)A(cl g*sg*s )BA(cl g*s  , then )))BA(int(clint(BA g*s  . Thus BA  is an s*g-  open set in YX  . By the same way, we can prove that A and B are s*g-  -open sets in X and Y respectevely if BA  is an s*g-  -open set in YX  . 3 . s*g- - Continuous Functions and s*g- - Irresolute Functions In this section , we introduce a new class of functions , namely , s*g-  -continuous functions and s*g-  -irresolute functions in topological spaces and study some of their properties. Definition(3.1): A function ),Y(),X(:f  is called s*g-  -continuous if )V(f 1 is an s*g-  -open set in X for every open set V in Y . Proposition(3.2): A function ),Y(),X(:f  is s*g-  -continuous iff )V(f 1 is an s*g-  - closed set in X for every closed set V in Y . Proof: It is Obvious . Proposition(3.3): Every continuous function is s*g-  -continuous . Proof: Follows from the definition (3.1) and the fact that every open set is s*g-  -open . Remark(3.4): The converse of Proposition (3.3) may not be true in general as shown in the following example: Example(3.5): Let }c,b,a{YX  , }}a{,,X{  & }}c,a{},a{,,Y{   },a{,,X{g*s   }}c,a{},b,a{ . Define ),Y(),X(:f  by : a)a(f  , b)b(f  & c)c(f   f is not continuous , but f is s*g-  -continuous , since X)Y(f 1  ,  )(f 1 , }c,a{})c,a({f 1  , and }a{})a({f 1  are s*g-  -open sets in X . Remark(3.6): s*g-continuous functions and s*g-  -continuous functions are in general independent . Consider the following examples:- 550 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Example(3.7): Let }c,b,a{YX  , },X{  & }}a{,,Y{    g*s and }}c,b{},c,a{},b,a{},c{},b{},a{,,X{g*s  . Define ),Y(),X(:f  by : a)a(f  , b)b(f  & c)c(f   f is s*g-continuous , but f is not s*g-  -continuous , since }a{ is open set in Y , but }a{})a({f 1  is not s*g-  -open in X . Also, in Example (3.5) f is s*g-  - continuous, but is not s*g-continuous , since }c,a{ is open set in Y , but }c,a{})c,a({f 1  is not s*g-open in X . Theorem(3.8): Every s*g-  -continuous function is  -continuous (resp. αg-continuous , gα- continuous , pre-continuous , b-continuous , -continuous) function . Proof: Follows from the Theorem (2.5) . Remark(3.9): The converse of Theorem (3.8) may not be true in general . Observe that in Example (3.7) f is pre-continuous (resp. b-continuous , -continuous , gα-continuous , αg- continuous ) function , but f is not s*g-  -continuous . Theorem(3.10): Every s*g-  -continuous function is semi-continuous function and gs- continuous function . Proof: Follows from the Theorem (2.8) . Remark(3.11): The converse of Theorem (3.10) may not be true in general as shown in the following example: Example(3.12): Let }c,b,a{YX  , }}b,a{},b{},a{,,X{  & }}c,a{},a{,,Y{  . Define ),Y(),X(:f  by : a)a(f  , b)b(f  & c)c(f   f is semi-continuous and gs-continuous, but f is not s*g-  -continuous, since }c,a{ is open in Y, but }c,a{})c,a({f 1  is not s*g-  -open in X , since  })))c,a(int({clint(}c,a{ g*s })))a({clint( g*s }a{})c,aint({  . Remark(3.13): Pre-continuous functions and  g-continuous functions are in general independent . Consider the following examples:- Example(3.14): Let }c,b,a{YX  , }}c,a{},a{,,X{  & }}b,a{},b{},a{,,Y{  . Define ),Y(),X(:f  by : a)a(f  , c)b(f  & b)c(f   f is  g-continuous , but f is not pre-continuous , since }b{ is open set in Y , but }c{})b({f 1  is not pre-open set in X , since }))c({clint(}c{  })c,bint({  . Example(3.15): Let  YX ,  usual topology & }}Q{,,{  . Define ),(),(:f  by : x)x(f  for each x  f is not  g-continuous , since Q is open in Y, but Q})Q({f 1  is not  g-open set in X . But f is pre-continuous . Remark(3.16): g-continuous functions and g  -continuous functions are in general independent . Consider the following examples:- Example(3.17): Let }c,b,a{YX  , }}a{,,X{  & }}b{,,Y{  . Define ),Y(),X(:f  by : a)a(f  , c)b(f  & b)c(f   f is g-continuous , but f is not g  - continuous, since }b{ is open set in Y, but }c{})b({f 1  is not g  -open set in X , since c}c{ }b,a{ is not g  -closed set in X . Example(3.18): Let }c,b,a{YX  , }}c,a{},a{,,X{  & }}b{,,Y{  . Define ),Y(),X(:f  by : b)a(f  , b)b(f  & a)c(f   f is g  -continuous , but f is not g-continuous, since }b{ is open set in Y, but }b,a{})b({f 1  is not g-open set in X , since }c{}b,a{ c  is not g-closed in X . 551 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 The following diagram shows the relationships between s*g-  -continuous functions and some other continuous functions: Proposition(3.19): If ),Y(),X(:f  is s*g-  -continuous, then ))A(f(cl))A(cl(f g*s  for every subset A of X . Proof: Since ))A(f(cl)A(f   )))A(f(cl(fA 1 . Since ))A(f(cl is a closed set in Y and f is s*g-  -continuous ,then by (3.2) )))A(f(cl(f 1 is an s*g-  -closed set in X containing A . Hence )))A(f(cl(f)A(cl 1g*s    . Therefore ))A(f(cl))A(cl(f g*s  . Theorem(3.20: Let ),Y(),X(:f  be a function . Then the following statements are equivalent:- i) f is s*g-  -continuous . ii) For each point x in X and each open set V in Y with V)x(f  , there is an s*g-  -open set U in X such that Ux  and V)U(f  . iii) For each subset A of X , ))A(f(cl))A(cl(f g*s  . iv) For each subset B of Y , ))B(cl(f))B(f(cl 11g*s    . Proof: )ii()i(  . Let YX:f  be an s*g-  -continuous function and V be an open set in Y s.t V)x(f  . To prove that , there is an s*g-  -open set U in X s.t Ux  and V)U(f  . Since f is s*g-  -continuous , then )V(f 1 is an s*g-  -open set in X s.t )V(fx 1 . Let )V(fU 1  V))V(f(f)U(f 1    V)U(f  . )i()ii(  . To prove that YX:f  is s*g-  -continuous . Let V be any open set in Y . To prove that )V(f 1 is an s*g-  -open set in X . Let )V(fx 1  V)x(f  .By hypothesis there is an s*g-  -open set U in X s.t Ux  and V)U(f   )V(fUx 1 . Thus by Theorem ((2.26),vii) )V(f 1 is an s*g-  -open set in X . Hence YX:f  is an s*g-  -continuous function . )iii()ii(  . Suppose that (ii) holds and let ))A(cl(fy g*s  and let V be any open neighborhood of y in Y . Since ))A(cl(fy g*s   )A(clx g*s  s.t y)x(f  . Since V)x(f  , then by (ii)  an s*g-  -open set U in X s.t Ux  and V)U(f  . Since )A(clx g*s  , then by Theorem ((2.26),viii) AU  and hence V)A(f  . Therefore we have ))A(f(cly  . Hence ))A(f(cl))A(cl(f g*s  . 552 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 )ii()iii(  . Let Xx  and V be any open set in Y containing )x(f . Let )V(fA c1  Ax  . Since cg*s V))A(f(cl))A(cl(f   A)V(f)A(cl c1 g*s    . Since Ax   )A(clx g*s  and by Theorem ((2.26),viii) there exists an s*g-  -open set U containing x such that AU  and hence V)A(f)U(f c  . )iv()iii(  . Suppose that (iii) holds and let B be any subset of Y . Replacing A by )B(f 1 we get from (iii) )B(cl)))B(f(f(cl)))B(f(cl(f 11g*s    . Hence ))B(cl(f))B(f(cl 11 g*s    . )iii()iv(  . Suppose that (iv) holds and let )A(fB  where A is a subset of X . Then we get from (iv) )))A(f(cl(f))A(f(f(cl)A(cl 11g*sg*s    . Therefore ))A(f(cl))A(cl(f g*s  . Definition(3.21): A function ),Y(),X(:f  is called s*g-  -irresolute if the inverse image of every s*g-  -open set in Y is an s*g-  -open set in X . Proposition(3.22): Every s*g-  -irresolute function is s*g-  -continuous . Proof: It is Obvious . Remark(3.23): The converse of Proposition (3.22) may not be true in general as shown in the following example: Example(3.24): Let }c,b,a{YX  , }}c,a{},c{},a{,,X{  & }}c,a{},a{,,Y{    g*s and }}c,a{},b,a{},a{,,Y{g*s   . Define ),Y(),X(:f  by : a)a(f  , b)b(f  & c)c(f   f is s*g-  -continuous, but f is not s*g-  -irresolute since }b,a{ is an s*g-  -open set in Y , but }b,a{})b,a({f 1  is not s*g-  -open set in X . Remark(3.25): continuous functions and s*g-  -irresolute functions are in general independent Consider the following examples:- Example(3.26 ): Let }c,b,a{YX  , }}c,b{},a{,,X{  & }}a{,,Y{  . Also , }}c,b{},a{,,X{g*s   & }}c,a{},b,a{},a{,,Y{g*s   . Define ),Y(),X(:f  by : a)a(f  , b)b(f  & c)c(f   f is continuous , but f is not s*g-  -irresolute , since }b,a{ is s*g-  -open set in Y , but }b,a{})b,a({f 1  is not s*g-  -open set in X . Example(3.27 ): Let }c,b,a{YX  , }}a{,,X{  & }}b,a{,,Y{  . Also , }}c,a{},b,a{},a{,,X{g*s   & }}b,a{,,Y{g*s   . Define ),Y(),X(:f  by : a)a(f  , b)b(f  & c)c(f   f is s*g-  -irresolute , but f is not continuous , Since }b,a{ is open in Y , but }b,a{})b,a({f 1  is not open in X . Theorem(3.28): Let ),Y(),X(:f  be a function . Then the following statements are equivalent:- (i) f is s*g-  -irresolute . (ii) For each Xx  and each s*g-  -neighborhood V of )x(f in Y, there is an s*g-  - neighborhood 553 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 U of x in X such that V)U(f  . (iii) The inverse image of every s*g-  -closed subset of Y is an s*g-  -closed subset of X . Proof: )ii()i(  . Let YX:f  be an s*g-  -irresolute function and V be an s*g-  - neighborhood of )x(f in Y . To prove that , there is an s*g-  -neighborhood U of x in X such that V)U(f  . Since f is an s*g-  -irresolute then , )V(f 1 is an s*g-  -neighborhood of x in X . Let )V(fU 1  V))V(f(f)U(f 1    V)U(f  . )i()ii(  . To prove that YX:f  is s*g-  -irresolute . Let V be an s*g-  -open set in Y . To prove that )V(f 1 is an s*g-  -open set in X . Let )V(fx 1  V)x(f   V is an s*g-  -neighborhood of )x(f .By hypothesis there is an s*g-  -neighborhood xU of x such that V)U(f x   )V(fU 1 x  , )V(fx 1   an s*g-  -open set xW of x such that )V(fUW 1xx  , )V(fx 1  )V(fW 1 )V(fx x 1      . Since  )V(fx 1 1 }x{)V(f     )V(fx x 1 W     )V(fx x 1 1 W)V(f     )V(f 1 is an s*g-  -open set in Y, since its a union of s*g-  -open sets . Thus YX:f  is an s*g-  -irresolute function . )iii()i(  . It is a obvious . Corollary(3.29): Let ),X( 11  and ),X( 22  be topological spaces . Then the projection functions 1211 XXX:  and 2212 XXX:  are s*g-  -irresolute functions . Proof: Let U be an s*g-  -open set in 1X , then 2 1 1 XU)U(   . Since U is s*g-  -open in 1X and 2X is s*g-  -open in 2X , then by Proposition (2.27) 2XU  is s*g-  -open in 21 XX  . Thus 1211 XXX:  is an s*g-  -irresolute function . Similaly we can prove that 2212 XXX:  is s*g-  -irresolute function . However the following theorem holds . The proof is easy and hence omitted . Theorem(3.30): If ),Y(),X(:f  and ),Z(),Y(:f  are functions, then:- i) If f and g are both s*g-  -irresolute functions , then so is fg  . ii) If f is s*g-  -irresolute and g is s*g-  -continuous , then fg  is s*g-  -continuous . iii) If f is s*g-  -continuous and g is continuous , then fg  is s*g-  -continuous . References 1 . Levine, N. (1963) Semi-open sets and semi-continuity in topological spaces , Amer. Math. Monthly, 70, 36-41 . 2 . Levine,N.(1970)Generalized closed sets in topology,Rend.Circ.Math.Palermo, 19 (2), 89-96 . 3 . Njasta, O. (1965) On some classes of nearly open sets , Pacific J. Math. 15, 961-970 . 4 . Mashhour, A.S. ; Abd El-Monsef,M.E. and El-Deeb,S.N. (1982) On precontinuous and weak precontinuous functions , Proc. Math. Phys. Soc. Egypt , 51, 47-53 . 5 . Andrijevic, D. (1996) On b-open sets, Mat. Vesnik , 48 (1-2) , 59-64 . 6 . 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I., (2014) Another Type Of Compactness In Bitopological Spaces , Journal of Al Rafidain University College , (to appear) . 555 | Mathematics 2014) عام 3العدد ( 27مجلة إبن الھيثم للعلوم الصرفة و التطبيقية المجلد Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 في الفضاءات التبولوجية  -S*g- المجموعات المفتوحة ولح صبيحة إبراھيم محمود سري طارق جمانة الجامعة المستنصرية / كلية العلوم / قسم الرياضيات 2014أيلول 1، قبل في 2014نيسان 9استلم في الخالصة -s*g- النمط من أسميناھا بالمجموعات المفتوحة من المجموعات اجديد ا ھذا البحث صنف قدمنا في -s*g المفتوحة من النمط و من ثم اثبتنا ان عائلة كل المجموعات الجزئية--s*g الفضاء التبولوجي من ),X(  تشكل تبولوجي على X الذي ھو انعم من . األساسية والخواص المكافئات كذلك درسنا هاستخدمنا ھذ ذلك عن فضال -. s*g-المغلقة من النمط والمجموعات -s*g-للمجموعات المفتوحة من النمط النمطمن بالدوال المستمرة صنف جديد من الدوال في الفضاءات التبولوجية أسميناه المجموعة في تعريف ودراسة - -s*g النمطمن والدوال المحيرة--s*g بعض خواص ھذه الدوال. قد درستو الدوال , -s*g-المجموعات المغلقة من النمط , s*g--المجموعات المفتوحة من النمط المفتاحية: الكلمات s*g .--الدوال المحيرة من النمط s*g ,--المستمرة من النمط