1 ON Weak g*sD -Sets And Associative Separation Axioms Sabiha I. Mahmood Department of Mathematics / College of Science / Al-Mustansiriyah University Received in:3 Septamber 2013 , Accepted in : 19 February 2014 Abstract In this paper, we introduce new classes of sets called g*sD -sets , g*sD −α -sets , g*spreD − - sets , g*sbD − -sets and g*sD −β -sets . Also, we study some of their properties and relations among them . Moreover, we use these sets to define and study some associative separation axioms . Keywords: s*g- iD -spaces , α -s*g- iD -spaces, pre-s*g- iD -space , b-s*g- iD -spaces, β -s*g- iD -spaces for 2,1,0i = . 351 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Introduction Tong, J . [1] , Calads,M. [2], Calads,M. and et.al. [3] , Jafari, S. [4] and Keskin,A. and Noiri, T. [5] introduced the notion of D-sets, sD -sets , αD -sets , preD -sets and bD -sets respectively by using open sets, semi-open sets, α-open sets, pre-open sets and b-open sets respectively and used the notion to define some associative separation axioms . Khan,M. and et.al.[6] introduced and investigated s*g-closed sets by using the concept of semi-open sets . In this paper we introduce and investigate new notions called g*sD -sets , g*sD −α -sets , g*spreD − - sets , g*sbD − -sets and g*sD −β -sets . Moreover, we use these notions to define some associative separation axioms . Recall that a subset A of a topological space ),X( τ is called semi-open [7] (resp. α -open [8], pre-open [9] , b-open[10] and β -open [11] ) set if ))A(int(clA ⊆ (resp . )))A(int(clint(A ⊆ , ))A(clint(A ⊆ , ))A(int(cl))A(clint(A ⊆ and )))A(cl(int(clA ⊆ ) . Also, a subset A of a topological space ),X( τ is called s*g-closed if U)A(cl ⊆ whenever UA ⊆ and U is semi-open in X [6] . The complement of an s*g-closed set is defined to be s*g-open . The family of all s*g-open subsets of ),X( τ is denoted by ),X(GO*S τ [6], this family from a topology on X which is finer than τ [6] . 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 and 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 [6] . A function ),Y(),X(:f σ→τ is called s*g- continuous [12] (resp. s*g-irresolute [12]) if the inverse image of every open ( resp. s*g-open) subset of Y is an s*g-open set in X . Throughout this paper ),X( τ and ),Y( σ (or simply X and Y) represent non- empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. Preliminaries First we recall the following definitions . Definition(1.1)[1]: A subset A of a topological space ),X( τ is called a D-set if there are two open sets U and V in X such that XU ≠ and V\UA = . Definition(1.2)[1]: A topological space ),X( τ is called a 0D -space if for any two distinct points x and y of X , there exists a D-set of X containing one of the points but not the other . Definition(1.3)[1]: A topological space ),X( τ is called a 1D -space if for any two distinct points x and y of X , there exists a D-set of X containing x but not y and a D-set of X containing y but not x . Definition(1.4)[1]: A topological space ),X( τ is called a 2D -space if for any two distinct points x and y of X , there are two D-sets U and V of X such that Ux ∈ , Vy ∈ and φ=VU  . 352 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Definition(1.5)[13]: A topological space ),X( τ is called a door space if each subset of X is either open or closed . Theorem(1.6): Let ),X( τ 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) )AX(int)A(clX g*sg*s −=− and )AX(cl)A(intX g*sg*s −=− . viii) )A(intx g*s∈ iff there is an s*g-open set U in X s.t AUx ⊆∈ . ix) )A(clx g*s∈ iff for every s*g-open set U containing x , φ≠AU  . x) )U(cl)U(cl g*sg*s  ∧∈α α ∧∈α α ⊆ and )U(int)U(int g*sg*s  ∧∈α α ∧∈α α ⊆ . Proof: It is a obvious . In this paper we introduce and investigate new notions called α -s*g-open sets , pre -s*g- open sets , b-s*g-open sets and β -s*g-open sets which are weaker than s*g-open . Moreover, we use these notions to define some associative separation axioms . 2. Weak Forms Of s*g-Open Sets In this section we introduce the following notions. Definitions(2.1): A subset A of a topological space ),X( τ is said to be : i) An α -s*g-open set if )))A((intcl(intA g*sg*s⊆ . ii) A pre-s*g-open set if ))A(cl(intA g*s⊆ . iii) A b-s*g-open set if ))A((intcl))A(cl(intA g*sg*s ⊆ . iv) Aβ -s*g-open set if )))A(cl((intclA g*s⊆ . Lemma(2.2): Let ),X( τ be a topological space , then the following properties hold: i) Every α -open (resp. pre-open, b-open, β -open) set is α -s*g-open (resp. pre-s*g-open, b- s*g- open, β -s*g-open) set . ii) Every s*g-open set is α -s*g-open . iii) Every α -s*g-open set is pre-s*g-open . iv) Every pre-s*g-open set is b-s*g-open . v) Every b-s*g-open set is β -s*g-open . Proof: It is obvious . Since every open set is s*g-open, then we have the following diagram for some types of open sets and s*g-open set . 353 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Figure No. (1): Relations between some types of open sets and s*g-open sets The converses need not be true in general as shown by the following examples . Example(2.3): Let }c,b,a{X = with the indiscrete topology },X{I φ==τ . Then }b,a{ is an s*g-open (resp. α -s*g-open) set, but it is not open (resp. not α -open) set . Example(2.4): Let }d,c,b,a{X = & }}d,b,a{},c,a{},a{,,X{ φ=τ . Then }d,c,a{ is an α -s*g- open set , but it is not s*g-open . Example(2.5): Let ℜ=X with the usual topology τ . Let QA = be the set of all rational numbers. Then A is an pre-s*g-open set (since A is pre-open set ) which is not α -s*g-open . Example(2.6): Let ℜ=X with the usual topology τ . Let ]1,0(A = . Then A is an b-s*g- open set (since A is b-open set ) which is not pre-s*g-open . Example(2.7): Let ℜ=X with the usual topology τ . Let ]1,0[QA = . Then A is a β -s*g open set (since A is β -open set ) which is not b-s*g-open . Theorem(2.8): If A is a pre-s*g-open subset of a topological space ),X( τ such that )U(clAU ⊆⊆ for a subset U of X , then U is an pre-s*g-open set . Proof: Since )U(clA ⊆ ⇒ )U(cl))U(cl(cl)A(cl =⊆ ⇒ ))U(cl(int))A(cl(int g*sg*s ⊆ . Since ))A(cl(intA g*s⊆ and AU ⊆ ⇒ ))U(cl(intU g*s⊆ . Thus U is an pre-s*g-open set . Theorem(2.9): A subset A of a topological space ),X( τ is semi-open if and only if A is β - s*g-open and ))A(int(cl))A(cl(int g*s ⊆ . Proof: Let A be semi-open, then ))A(int(clA ⊆ )))A(cl((intcl g*s⊆ and hence A is β -s*g- open. Also, since ))A(int(clA ⊆ ⇒ ))A(int(cl)A(cl ⊆ ⇒ ))A(int(cl))A(cl(int g*s ⊆ . Conversely, let A be β -s*g-open and ))A(int(cl))A(cl(int g*s ⊆ . Then )))A(int(cl(cl)))A(cl((intclA g*s ⊆⊆ ))A(int(cl= and hence A is semi-open . Lemma(2.10)[13]: Let ),X( τ be a topological space . If U is an open set in X, then )AU(cl)A(clU  ⊆ for any subset A of X . Propositions(2.11): Let ),X( τ be a topological space , then: i) The intersection of a pre-s*g-open set and an open set is pre-s*g-open . ii) The intersection of aβ -s*g-open set and an open set is β -s*g-open . 354 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 iii) The intersection of a b-s*g-open set and an open set is b-s*g-open . iv) The intersection of an α -s*g-open set and an open set is α -s*g-open . Proof: We prove only the first case since the other cases are similarly shown . i) Let A be a pre-s*g-open set and U be an open set in X . Since every open set is s*g-open , then ))A(cl(intA g*s⊆ and )U(intU g*s= . By Lemma (2.10), we have ))A(cl(int)U(intAU g*sg*s  ⊆ ))A(clU(int g*s = ))AU(cl(int g*s ⊆ . Therefore UA  is pre-s*g-open . Remark(2.12): We note that the intersection of two pre-s*g-open (resp. b-s*g-open,β -s*g- open, α -s*g-open) sets need not be pre-s*g-open (resp. b-s*g-open, β -s*g-open, α -s*g- open) as can be seen from the following examples: Example(2.13): Let ℜ=X with the usual topology τ . Let QA = and }1{QB c = , then A and B are pre-s*g-open , but }1{BA = which is not β -s*g-open since })))1({cl((intcl g*s }))1({(intcl g*s= φ=φ= })({cl . Example(2.14): Let }c,b,a{X = & }}c,b{,,X{ φ=τ . Then }b,a{ and }c,a{ are α -s*g-open sets, but }a{}c,a{}b,a{ = is not α -s*g-open . Theorem(2.15): If }:A{ ∧∈αα is a collection of b-s*g-open (resp. pre-s*g-open, β -s*g- open, α -s*g-open) sets of a topological space ),X( τ , then  ∧∈α αA is b-s*g-open (resp. pre-s*g-open, β - s*g-open, α -s*g-open). Proof: We prove only the first case since the other cases are similarly shown . Since ))A((intcl))A(cl(intA g*sg*s ααα ⊆  for every ∧∈α , we have: ))]A((intcl))A(cl([intA g*sg*s αα ∧∈α∧∈α α ⊆  ))]A((intcl[))]A(cl(int[ g*sg*s   ∧∈α αα ∧∈α = ))]A(int(cl[]))A(cl([int g*sg*s   ∧∈α α ∧∈α α⊆ ))]A((intcl[]))A(cl([int g*sg*s   ∧∈α α ∧∈α α⊆ Therefore  ∧∈α αA is b-s*g-open . Proposition(2.16): Let ),X( τ be a topological space and XA ⊆ . If A is a b-s*g-open set such that φ=)A(int g*s , then A is pre-s*g-open . Proof: Since A is b-s*g-open, then ))A((intcl))A(cl(intA g*sg*s ⊆ . Since φ=)A(int g*s , then φ=))A((intcl g*s , therefore ))A(cl(intA g*s⊆ . Thus A is a pre-s*g-open set . Propositions(2.17): If ),X( τ is a door space , then: i) Every pre-s*g-open set is s*g-open . ii) Everyβ -s*g-open set is b-s*g-open . 355 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Proof: i) Let A be an pre-s*g-open set. If A is open, then A is s*g-open. Otherwise, A is closed and hence ))A(cl(intA g*s⊆ )A(int g*s= . Therefore, )A(intA g*s= and thus A is an s*g-open set. ii) Let A be anβ -s*g-open set . If A is open , then A is b-s*g-open . Otherwise , A is closed and hence )))A(cl((intclA g*s⊆ ))A((intcl g*s= ))A((intcl))A(cl(int g*sg*s ⊆ . Therefore A is an b-s*g-open set . Definitions(2.18): A subset A of a topological space ),X( τ is called: i) An s*g-t-set if ))A(cl(int)Aint( g*s= . ii) An s*g-B-set if VUA = , where τ∈U and V is an s*g-t-set . Proposition(2.19): Let A and B be subsets of a topological space ),X( τ . If A and B are s*g- t-sets, then BA  is an s*g-t-set . Proof: Let A and B be s*g-t-sets . Then we have: ))BA(cl(int g*s  ))B(cl)A(cl(int g*s ⊆ ))B(cl(int))A(cl(int g*sg*s = )Bint()Aint( = )BAint( = . Since ))BA(cl(int)BAint( g*s  ⊆ , then ))BA(cl(int)BAint( g*s  = and hence BA  is an s*g-t-set . From the following example one can deduce that a pre-s*g-open set and a s*g-B-set are independent . Example(2.20): Let X = R with the usual topology τ . Then Q\R is pre-s*g-open, but it is not an s*g-B-set (since Q\RRQ\R = , where τ∈R ,but Q\R is not an s*g-t-set ) and (0,1] is an s*g-B-set ( since ]1,0(R]1,0( = , where τ∈R and (0,1] is an s*g-t-set ) which is not pre-s*g-open (since )1,0(]))1,0((cl(int]1,0( g*s =⊄ ) . Proposition(2.21): Let ),X( τ be a topological space and XA ⊆ . Then the following are equivalent: i) A is open . ii) A is pre-s*g-open and an s*g-B-set . Proof: )ii()i( ⇒ . Let A be open. Then )A(intA g*s= ))A(cl(int g*s⊆ and A is pre-s*g-open . Also , XAA = , where τ∈A and X is an s*g-t-set and hence A is an s*g-B-set . )i()ii( ⇒ . Since A is an s*g-B-set, we have VUA = , where τ∈U and V is an s*g-t-set . By the hypothesis, A is also pre-s*g-open and we have: ))A(cl(intA g*s⊆ ))VU(cl(int g*s = ))V(cl)U(cl(int g*s ⊆ ))V(cl(int))U(cl(int g*sg*s = )Vint())U(cl(int g*s = Hence U)VU(VUA  == U))Vint())U(cl((int g*s ⊆ )Vint()U))U(cl((int g*s = )Vint(U = )Vint()Uint( = )VUint( = )Aint(= . Therefore )Aint(A = and A is open . 356 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Definitions(2.22): A subset A of a topological space ),X( τ is called : i) An s*g- αt -set if )))A((intcl(int)Aint( g*sg*s= . ii) An s*g- αB -set if VUA = , where τ∈U and V is an s*g- αt -set . Proposition(2.23): Let A and B be subsets of a topological space ),X( τ . If A and B are s*g- αt -sets, then BA  is an s*g- αt -set . Proof: Let A and B be s*g- αt -sets . Then we have: ))BA((intcl(int g*sg*s  )))B(int)A((intcl(int g*sg*sg*s = )))B((intcl)A((intcl(int g*sg*sg*s ⊆ )))B((intcl(int)))A((intcl(int g*sg*sg*sg*s = )Bint()Aint( = )BAint( = . Since )))BA((intcl(int)BAint( g*sg*s  ⊆ , then ))BA((intcl(int)BAint( g*sg*s  = and hence BA  is an s*g- αt -set . From the following example one can deduce that an α -s*g-open set and an s*g- αB -set are independent . Example(2.24): Let X = R with the usual topology τ . Then (0,1] is an s*g- αB -set which is not α -s*g-open . Also, in Example (2.3), }b,a{A = is an α -s*g-open set, but is not an s*g- αB -set. Proposition(2.25): Let ),X( τ be a topological space and XA ⊆ . Then the following are equivalent: i) A is open . ii) A is α -s*g-open and an s*g- αB -set . Proof: )ii()i( ⇒ . Let A be open. Then )A(intA g*s= ))A((intcl g*s⊆ and ))A((intcl(intA g*sg*s⊆ Therefore A is α -s*g-open. Also, XAA = , where τ∈A and X is an s*g- αt -set and hence A is an s*g- αB -set . )i()ii( ⇒ . Since A is an s*g- αB -set, we have VUA = , where τ∈U and V is an s*g- αt - set . By the hypothesis, A is also α -s*g-open, and we have: )))A((intcl(intA g*sg*s⊆ )))VU((intcl(int g*sg*s = )))V(int)U((intcl(int g*sg*sg*s = )))V((intcl)U((intcl(int g*sg*sg*s ⊆ )))V((intcl(int)U((intcl(int g*sg*sg*sg*s = )Vint())U(cl(int g*s ⊆ Hence U)VU(VUA  == U))Vint())U(cl((int g*s ⊆ )Vint()U))U(cl((int g*s = )Vint(U = )Vint()Uint( = )VUint( = )Aint(= . Therefore )Aint(A = and A is open . 357 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Definition(2.26): A subset A of a topological space ),X( τ is called an s*g-set if VUA = , where τ∈U and )V(int)Vint( g*s= . From the following example one can deduce that an s*g-open set and an s*g-set are independent . Example(2.27): Let X = R with the usual topology τ . Then Q)1,0(A = is an s*g-set which is not s*g-open . Also, in Example (2.3) , }b,a{A = is an s*g-open set, but is not an s*g-set . Proposition(2.28): Let ),X( τ be a topological space and XA ⊆ . Then the following are equivalent: i) A is open . ii) A is s*g-open and an s*g-set . Proof: )ii()i( ⇒ . This is obvious . )i()ii( ⇒ . Since A is an s*g-set, we have VUA = , where τ∈U and )V(int)Vint( g*s= . By the hypothesis, A is also s*g-open and we have: )A(intA g*s= )VU(int g*s = )V(int)U(int g*sg*s = )Vint(U = )Vint()Uint( = )VUint( = )Aint(= . Therefore A is open . Definitions(2.29): A topological space ),X( τ is said to satisfy: i) The s*g-condition if every s*g-open set is s*g-t-set . ii) The s*g- αB -condition if every α -s*g-open set is s*g- αB -set . iii) The s*g-B-condition if every pre-s*g-open set is s*g-B-set . Definition(2.30): A topological space ),X( τ is called an s*g- 0T -space [14] (resp. α -s*g- 0T - space, pre-s*g- 0T -space, b-s*g- 0T -space, β -s*g- 0T -space) if for any two distinct points x and y of X , there exists an s*g-open (resp. α -s*g-open, pre-s*g-open, b-s*g-open, β -s*g- open) set of X containing one of the points but not the other . Definition(2.31): A topological space ),X( τ is called an s*g- 1T -space [14] (resp. α -s*g- 1T - space, pre-s*g- 1T -space, b-s*g- 1T -space, β -s*g- 1T -space) if for any two distinct points x and y of X , there exists an s*g-open (resp. α -s*g-open, pre-s*g-open, b-s*g-open, β -s*g-open) set of X containing x but not y and an s*g-open (resp. α -s*g-open, pre-s*g-open, b-s*g-open, β -s*g-open) set of X containing y but not x . Definition(2.32): A topological space ),X( τ is called an s*g- 2T -space [14] (resp. α -s*g- 2T - space, pre-s*g- 2T -space, b-s*g- 2T -space, β -s*g- 2T -space) if for any two distinct points x and y of X , there are two s*g-open (resp. α -s*g-open, pre-s*g-open, b-s*g-open, β -s*g- open) sets U and V of X such that Ux ∈ , Vy ∈ and φ=VU  . 3. Weak g*sD -Sets And Associative Separation Axioms 358 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 In this section we introduce and investigate new notions called g*sD -sets , g*sD −α -sets , g*spre D − -sets , g*sbD − -sets and g*sD −β -sets and we use these notions to define and study some associative separation axioms . Definition(3.1): A subset A of a topological space ),X( τ is called an g*sD -set (resp. g*sD −α - set, g*spre D − -set, g*sbD − -set, g*sD −β -set) if there are two s*g-open (resp. α -s*g-open, pre-s*g- open, b-s*g-open, β -s*g-open) sets U and V in X such that XU ≠ and V\UA = . Remark(3.2): In definition (3.1), if XU ≠ and φ=V , then every proper s*g-open (resp. α - s*g-open, pre-s*g-open, b-s*g-open,β -s*g-open) subset U of X is an g*sD -set (resp. g*sD −α - set , g*spre D − -set, g*sbD − -set, g*sD −β -set) . Proposition(3.3): In any topological space ),X( τ . i) Any D-set is g*sD -set . ii) Any g*sD -set is g*sD −α -set . iii) Any g*sD −α -set is g*spreD − -set . iv) Any g*spre D − -set is g*sbD − -set . v) Any g*sbD − -set is g*sD −β -set . Proof: Follows from Lemma (2.2) . Proposition(3.4): In any door space ),X( τ . i) Any g*spre D − -set is g*sD -set . ii) Any g*s D −β -set is g*sbD − -set . Proof: Follows from Proposition (2.17) . Proposition(3.5): In any topological space satisfies s*g-condition any g*sD -set is D-set . Proof: Suppose that A is an g*sD -set , then there are two s*g-open sets U and V in X such that XU ≠ and V\UA = . Hence ))U(cl(int)U(intU g*sg*s ⊆= and ))V(cl(int)V(intV g*sg*s ⊆= . Since X is satisfy the s*g-condition, then U and V are s*g-t- sets . Therefore )Uint(U ⊆ and )Vint(V ⊆ . Hence U and V are open-sets . Thus A is D-set . Proposition(3.6): In any topological space satisfies s*g- αB -condition any g*sD −α -set is D-set . Proof: Follows from Proposition (2.25) . 359 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Proposition(3.7): In any topological space satisfies s*g-B-condition any g*spre D − -set is D-set . Proof: Follows from Proposition (2.21) . From above propositions we can get the following diagram . (1) s*g-Bα-Condition (2) s*g-Condition (3) s*g-B-Condition (4) door space Figure No. (2): Relations among the weak g*sD - sets Definition(3.8): A function ),Y(),X(:f σ→τ is said to be α -s*g-continuous (resp. pre-s*g- continuous, b-s*g-continuous, β -s*g-continuous) if )V(f 1− is α -s*g-open (resp. pre-s*g- open, b-s*g-open, β -s*g-open) set in X for each open set V in Y . Definition(3.9): A function ),Y(),X(:f σ→τ is said to be α -s*g-irresolute ( resp. pre- s*g- irresolute ,b-s*g-irresolute,β -s*g-irresolute) if )V(f 1− is α -s*g-open (resp. pre-s*g-open, b- s*g-open, β -s*g-open) set in X for each α -s*g-open (resp. pre-s*g-open, b-s*g-open, β -s*g- open) set V in Y . Theorem(3.10): If ),Y(),X(:f σ→τ is an α -s*g-continuous (resp. s*g-continuous, pre- s*g-continuous, b-s*g-continuous, β -s*g-continuous) surjective function and S is a D-set in Y, then the inverse image of S is an g*sD −α -set (resp. g*sD -set, g*spreD − -set, g*sbD − -set, g*sD −β - set) in X . Proof: Let S be a D-set in Y , then there are two open sets 1U and 2U in Y such that 21 U\US = and YU1 ≠ . Since f is α -s*g-continuous ,then )U(f 1 1− and )U(f 2 1− are α -s*g- open sets in X. Since YU1 ≠ and f is surjective , then X)U(f 1 1 ≠− . Hence )U(f\)U(f)S(f 2 1 1 11 −−− = is a g*sD −α -set in X . By the same way we can prove that other cases . Theorem(3.11): If ),Y(),X(:f σ→τ is an α -s*g-irresolute (resp. s*g-irresolute , pre- s*g-irresolute , b-s*g-irresolute ,β -s*g-irresolute) surjective function and S is an g*sD −α - set ( resp. 360 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 g*sD -set , g*spreD − -set , g*sbD − -set , g*sD −β -set) in Y, then the inverse image of S is an g*sD −α - set (resp. g*sD -set , g*spreD − -set , g*sbD − -set , g*sD −β -set) in X . Proof: Let S be an g*sD −α -set in Y , then there are two α -s*g-open sets 1U and 2U in Y such that 21 U\US = and YU1 ≠ . Since f is α -s*g- irresolute , then )U(f 1 1− and )U(f 2 1− are α -s*g-open sets in X . Since YU1 ≠ and f is surjective , then X)U(f 1 1 ≠− . Hence )U(f\)U(f)S(f 2 1 1 11 −−− = is an g*sD −α -set in X. By the same way we can prove that other cases. Definitions(3.12): A topological space ),X( τ is called: (i) An s*g- 0D -space (resp. α -s*g- 0D -space, pre-s*g- 0D -space, b-s*g- 0D -space, β -s*g- 0D -space) if for any two distinct points x and y of X , there exists an g*sD -set (resp. g*sD −α -set, g*spre D − -set , g*sbD − -set, g*sD −β -set) of X containing one of the points but not the other . (ii) An s*g- 1D -space (resp. α -s*g- 1D -space, pre-s*g- 1D -space, b-s*g- 1D -space, β -s*g- 1D - space) if for any two distinct points x and y of X , there exists an g*sD -set (resp. g*sD −α -set, g*spre D − -set , g*sbD − -set, g*sD −β -set) of X containing x but not y and an g*sD -set (resp. g*sD −α - set, g*spre D − -set , g*sbD − -set, g*sD −β -set) of X containing y but not x . (iii) An s*g- 2D -space (resp. α -s*g- 2D - space, pre-s*g- 2D -space, b-s*g- 2D -space, β -s*g- 2D -space) if for any two distinct points x and y of X , there are two g*sD -sets (resp. g*sD −α - sets, g*spre D − -sets, g*sbD − -sets, g*sD −β -sets) U and V of X such that Ux ∈ , Vy ∈ and φ=VU  . Theorem(3.13): (i) Every s*g- iT -space (resp. α -s*g- iT -space , pre-s*g- iT -space , b-s*g- iT -space, β -s*g- iT -space) is s*g- 1iT − -space (resp. α -s*g- 1iT − -space, pre-s*g- 1iT − -space, b- s*g- 1iT − -space, β -s*g - 1iT − -space) , 2,1i = . (ii) Every s*g- iT -space (resp. α -s*g- iT -space , pre-s*g- iT -space , b-s*g- iT -space , β -s*g- iT -space) is s*g- iD -space (resp. α -s*g- iD -space, pre-s*g- iD -space, b-s*g- iD -space, β - s*g- iD -space) , 2,1,0i = . (iii) Every s*g- iD -space (resp. α -s*g- iD -space , pre-s*g- iD -space , b-s*g- iD -space ,β - s*g- iD -space) is s*g- 1iD − -space (resp. α -s*g- 1iD − -space, pre-s*g- 1iD − -space, b-s*g- 1iD − - space, β -s*g - 1iD − -space) , 2,1i = . Proof: (i) It is obvious . (ii) Follows from Remark (3.2). (iii) It is obvious . Remark(3.14): The converse of theorem (3.13), no. (i) may not be true . Consider the following examples: 361 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Example(3.15): Let X be any infinte set and let τ ={ cU:XU ⊆ is finite} }{φ . Then ),X( τ is an s*g- 1T -space (resp. α -s*g- 1T -space , pre-s*g- 1T -space , b-s*g- 1T -space , β -s*g- 1T -space) , but is not an s*g- 2T -space (resp. α -s*g- 2T -space , pre-s*g- 2T -space , b-s*g- 2T -space , β -s*g- 2T -space) . Example(3.16): Let }b,a{X = and }}a{,,X{ φ=τ . Then ),X( τ is s*g- 0T -space (resp. α -s*g- 0T - space, pre-s*g- 0T -space, b-s*g- 0T -space, β -s*g- 0T -space), but is not s*g- 1T -space (resp. α - s*g- 1T -space, pre-s*g- 1T -space, b-s*g- 1T -space, β -s*g- 1T -space) . Remark(3.17): The converse of theorem (3.13), no.(ii) may not be true . Consider the following examples: Example(3.18): Let }c,b,a{X = and }}c,a{},b,a{},a{,,X{ φ=τ . Then s*g-open sets in X = open sets in X . Hence ),X( τ is s*g- iD -space , but is not s*g- iT -space , 2,1i = . Example(3.19): Let }c,b,a{X = and }}a{,,X{ φ=τ . Then α -s*g-open sets in X = pre-s*g- open sets in X = b-s*g-open sets in X = β -s*g-open sets in X = }}c,a{},b,a{},a{,,X{ φ . Hence ),X( τ is α -s*g- iD -space (resp. pre-s*g- iD -space , b-s*g- iD -space ,β -s*g- iD -space), but is not α -s*g- iT -space (resp. pre-s*g- iT -space , b-s*g- iT -space , β -s*g- iT -space) , 2,1i = . Remark(3.20): The converse of theorem (3.13), no. (iii) may not be true . In example (3.16) , ),X( τ is s*g- 0D -space (resp. α -s*g- 0D - space, pre-s*g- 0D -space, b-s*g- 0D -space, β -s*g- 0D -space) , but is not s*g- 1D -space (resp. α -s*g- 1D -space, pre-s*g- 1D -space, b-s*g- 1D - space, β -s*g- 1D -space) . Theorem(3.21): A topological space ),X( τ is an α -s*g- 0D -space (resp. s*g- 0D -space, pre-s*g- 0D -space, b-s*g- 0D -space, β -s*g- 0D -space) if and only if it is an α -s*g- 0T -space (resp. s*g- 0T - space, pre-s*g- 0T -space, b-s*g- 0T -space, β -s*g- 0T -space) . Proof: Sufficiency. Follows from Theorem (3.13), no. (ii) . Necessity. Let Xy,x ∈ such that yx ≠ . Since ),X( τ is α -s*g- 0D -space, then there exists an g*sD −α -set U such that Ux ∈ , Uy ∉ . Let 21 P\PU = , where XP1 ≠ and 21 P,P are α -s*g- open sets in X . By Uy ∉ we have two cases: (i) 1Py ∉ (ii) 1Py ∈ and 2Py ∈ . In case (i) 1Py ∉ and 21 P\PUx =∈ ⇒ 1Px ∈ and 1Py ∉ . In case (ii) 1Py ∈ and 2Py ∈ and 21 P\Px ∈ ⇒ 2Px ∉ ⇒ 2Py ∈ and 2Px ∉ . Thus in both cases, we obtain that ),X( τ is an α -s*g- 0T -space . By the same way we can prove that other cases . Theorem(3.22): A topological space ),X( τ is an α -s*g- 1D -space (resp. s*g- 1D -space, pre-s*g- 1D -space, b-s*g- 1D -space, β -s*g- 1D -space) if and only if it is an α -s*g- 2D -space (resp. s*g- 2D - space, pre-s*g- 2D -space, b-s*g- 2D -space, β -s*g- 2D -space) . 362 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Proof: Sufficiency . Follows from Theorem (3.13) , no. (iii) . Necessity . Let Xy,x ∈ such that yx ≠ . Since ),X( τ is an α -s*g- 1D -space, then there exists g*sD −α -sets U and V in X such Ux ∈ , Uy ∉ and Vy ∈ , Vx ∉ . Let 21 P\PU = and 43 P\PV = ,where 4321 P,P,P,P are α -s*g-open sets in X and XP1 ≠ and XP3 ≠ . By Vx ∉ we have two cases : (i) 3Px ∉ (ii) 3Px ∈ and 4Px ∈ . In case (i): 3Px ∉ . By Uy ∉ we have two subcases: (a) 1Py ∈ and 2Py ∈ (b) 1Py ∉ . Subcase (a): 1Py ∈ and 2Py ∈ . We have 21 P\Px ∈ , 2Py ∈ and φ=221 P)P\P(  . Observe that XP2 ≠ since φ≠U , thus by Remark (3.2) 2P is an g*sD −α -set . Subcase (b): 1Py ∉ . Since 21 P\Px ∈ and 3Px ∉ , then )PP(\Px 321 ∈ and since 43 P\Py ∈ and 1Py ∉ , then )PP(\Py 143 ∈ . Observe also from theorem (2.15) that )PP( 32  and )PP( 14  are α -s*g-open sets . Hence )PP(\Px 321 ∈ , )PP(\Py 143 ∈ and φ=))PP(\P())PP(\P( 143321  . In case (ii): 3Px ∈ and 4Px ∈ . We have 43 P\Py ∈ , 4Px ∈ and φ=443 P)P\P(  . Observe that XP4 ≠ since φ≠V , thus by Remark (3.2) 4P is an g*sD −α -set . Hence ),X( τ is an α -s*g- 2D -space . By the same way we can prove that other cases . Corollary(3.23): If ),X( τ is an α -s*g- 1D -space (resp. s*g- 1D -space, pre-s*g- 1D -space, b- s*g- 1D -space, β -s*g- 1D -space) ,then it is an α -s*g- 0T -space (resp. s*g- 0T -space, pre-s*g- 0T -space , b-s*g- 0T -space, β -s*g- 0T -space) . Proof: Follows from Theorem (3.13) , no. (iii) and Theorem (3.21) . Remark(3.24): The converse of Corollary (3.23) may not be true . In example (3.16), ),X( τ is α -s*g- 0T -space (resp. s*g- 0T -space, pre-s*g- 0T -space, b-s*g- 0T -space, β -s*g- 0T -space), but is not an α -s*g- 1D -space (resp. s*g- 1D -space, pre-s*g- 1D -space, b-s*g- 1D -space, β - s*g- 1D -space) . Propositions(3.25): (i) Every s*g- iD -space is α -s*g- iD -space , 2,1,0i = . (ii) Every α -s*g- iD -space is pre-s*g- iD -space , 2,1,0i = . (iii) Every pre-s*g- iD -space is b-s*g- iD -space , 2,1,0i = . (iv) Every b-s*g- iD -space is β -s*g- iD -space , 2,1,0i = . Remark(3.26): The converse of proposition (3.25), no.(i) may not be true . In example (3.19), ),X( τ is α -s*g- iD -space , but is not s*g- iD -space , 2,1,0i = . Remark(3.27): The converse of proposition (3.25), no. (iii) may not be true . Consider the following example: 363 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Example(3.28): Let }c,b,a{X = & }}b,a{},b{},a{,,X{ φ=τ . Then pre-s*g-open sets in X = open sets in X and b-s*g-open sets in X= }}c,b{},c,a}{b,a{},b{},a{,,X{ φ . Hence ),X( τ is b- s*g- iD -space , but is not pre-s*g- iD -space , 2,1i = . From above we can get the following diagram . Figure No. (3): Relations among the types of separation axioms Definition(3.29): A subset A of a topological space ),X( τ is called an α -s*g-neighborhood (resp. s*g-neighborhood, pre-s*g-neighborhood, b-s*g-neighborhood,β -s*g-neighborhood) of a point x in X if there exists an α -s*g-open (resp. s*g-open, pre-s*g-open, b-s*g-open,β -s*g- open) set U in X such that AUx ⊆∈ . Definition(3.30): Let ),X( τ be a topological space . A point Xx ∈ which has X as the only α - s*g-neighborhood (resp. s*g-neighborhood , pre-s*g-neighborhood , b-s*g-neighborhood ,β - s*g-neighborhood) is called an α -s*g-neat (resp. s*g-neat, pre-s*g-neat, b-s*g-neat,β -s*g- neat) point . Theorem(3.31): For an α -s*g- 0T -space (resp. s*g- 0T -space , pre-s*g- 0T -space , b-s*g- 0T -space , β -s*g- 0T -space) ),X( τ the following are equivalent . (i) ),X( τ is an α -s*g- 1D -space (resp. s*g- 1D -space, pre-s*g- 1D -space, b-s*g- 1D -space, β - s*g- 1D -space) . (ii) ),X( τ has no α -s*g-neat (resp. s*g-neat, pre-s*g-neat, b-s*g-neat, β -s*g-neat) point . Proof: )ii()i( ⇒ . Since ),X( τ is an α -s*g- 1D -space , then each point x of X is contained in a g*sD −α -set V\UG = , where U and V are α -s*g-open sets and thus in U . By definition XU ≠ . This implies that x is not an α -s*g-neat point . )i()ii( ⇒ . If ),X( τ is an α -s*g- 0T -space , then for each distinct points Xy,x ∈ , at least one of them, say x has an α -s*g-neighborhood U containing x , but not y . Thus U is different 364 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 from X and therefore by Remark (3.2), U is an g*sD −α -set . Since X has no α -s*g-neat point, then y is not an α -s*g-neat point . Thus there exists an α -s*g-neighborhood V of y such that XV ≠ . Therefore , U\Vy ∈ , U\Vx ∉ and U\V is an g*sD −α -set . Hence ),X( τ is an α -s*g- 1D - space. Theorem(3.32): Let ),Y(),X(:f σ→τ be an α -s*g-continuous (resp. s*g-continuous , pre-s*g-continuous, b-s*g-continuous, β -s*g-continuous) bijective function . If ),Y( σ is a iD - space , then ),X( τ is an α -s*g- iD -space (resp. s*g- iD -space, pre-s*g- iD -space, b-s*g- iD -space, β -s*g- iD -space), 2,1,0i = . Proof: Suppose that ),Y( σ is a 2D -space . Let Xy,x ∈ such that yx ≠ . Since f is injective and Y is a 2D -space , then there exists disjoint D-sets 1G and 2G of Y such that 1G)x(f ∈ and 2G)y(f ∈ . By Theorem (3.10), )G(f 1 1− and )G(f 2 1− are g*sD −α -sets in X such that )G(fx 1 1−∈ , )G(fy 2 1−∈ and φ=−− )G(f)G(f 2 1 1 1  . Hence ),X( τ is an α -s*g- 2D -space . Theorem(3.33): Let ),Y(),X(:f σ→τ be an α -s*g-irresolute (resp. s*g-irresolute , pre- s*g- irresolute, b-s*g-irresolute, β -s*g-irresolute) bijective function . If ),Y( σ is an α -s*g- iD - space (resp. s*g- iD -space, pre-s*g- iD -space, b-s*g- iD -space, β -s*g- iD -space) , then ),X( τ is an α -s*g- iD -space (resp. s*g- iD -space , pre-s*g- iD -space , b-s*g- iD -space , β -s*g- iD - space ) , 2,1,0i = . Proof: Suppose that ),Y( σ is an α -s*g- 2D -space . Let Xy,x ∈ such that yx ≠ . Since f is injective and Y is an α -s*g- 2D -space , then there exists disjoint g*sD −α -sets 1G and 2G of Y such that 1G)x(f ∈ and 2G)y(f ∈ . By Theorem (3.11) , )G(f 1 1− and )G(f 2 1− are g*sD −α - sets in X such that )G(fx 1 1−∈ , )G(fy 2 1−∈ and φ=−− )G(f)G(f 2 1 1 1  . Hence ),X( τ is an α -s*g- 2D -space . 365 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 References 1. Tong, J. (1982) A separation axioms between 0T and 1T , Ann. Soc. Sci. Bruxelles 96 II,85- 90 . 2. Calads,M. (1997) A Separation axioms between semi- 0T and semi- 1T , Mem. Fac. Sci. Kochi Univ. Ser. A Math. 181,37-42. 3. Calads,M. ; Georgiou, D.N. and Jafari, S. (2003) Characterization of low separation axioms via α -open sets and α -closure operator , Bol. Soc. Paran. Mat.(3s), 21(1/2), 1-14 . 4. Jafari, S. On a weak separation axioms ,Far East J. Math. Sci. (to appear) . 5. Keskin,A. and Noiri, T. (2009) On bD-sets and associated separation axioms , Bulletin of the Iranian Math. Soc. 35,1,179-198. 6. Khan,M. ; Noiri, T . and Hussain, M .(2008) On s*g-closed sets and s*-normal spaces , 48 , 31-41 . 7. Levine,N. (1963) Semi-open sets and semi-continuity in topological spaces , Amer. Math. Monthly 70, 36-41 . 8. Njastad,O. (1965) On some classes of nearly open sets , Pacific J. Math. 15, 961-970 . 9. 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 . 10. Andrijevic, D. (1996) On b-open sets, Mat. Vesnik , 48 (1-2) , 59-64 . 11. Abd El-Monsef,M.E. ; El-Deeb , S.N. and Mahmoud,R.A. (1983) β -open sets andβ - continuous Mappings , Bull. Fac. Sci. Assuit Univ. 12, 77-90 . 12. Veerakumar,M.K.R.S. (2001) ĝ -closed sets and G L̂ C-functions , Indian J.Math., 43,2 , 231-247. 13. Kelly,J.L.(1955) General Topology , VAN NOSTRAND , New York . 14. S.I. and Afrah, M . (2010) S*-separation axioms , Iraqi Journal of Science, University of Baghdad, 51, 1 ,145-153 . 366 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 صل المشتقة منھافال وبدیھیات الضعیفة g*sD-مجموعات حول ال صبیحة إبراھیم محمود المستنصریةالجامعة -كلیة العلوم -قسم الریاضیات 2014شباط 19 :، قبل البحث في 2013ایلول 3أستلم البحث في : الخالصة -المجموعات, g*sD -بالمجموعات اسمیناھا جدیدة من المجموعات اصناف یمقدبت قمنا في ھذا البحث g*sD −α g*spreD -المجموعات , − g*sbD -المجموعات , g*sD -والمجموعات − −βھذه خواص بعض درسنا . كذلك بعض بدیھیات الفصلوالعالقات بینھم . فضال عن ذلك استخدمنا ھذه المجموعات في تعریف ودراسھ المجموعات . المشتقة منھا iD -الفضاءات : المفتاحیة الكلمات s*g- ,الفضاءات- iD -s*g- α , الفضاءات- iD pre-s*g- ,الفضاءات- iD b- s*g- , الفضاءات- iD -s*g-β )2,1,0i =(. 367 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Sabiha I. Mahmood Department of Mathematics / College of Science / Al-Mustansiriyah University صبيحة إبراهيم محمود قسم الرياضيات - كلية العلوم- الجامعة المستنصرية أستلم البحث في :3 ايلول 2013 , قبل البحث في: 19 شباط 2014 الخلاصة