2009) 3( 22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد قبل المفتوحة شبه المجموعاتبعض النتائج حول احمد إبراهیم ناصررشا ناصر مجید ، د الهیثم ،أبن -كلیة التربیة سم الریاضیات،ق جامعة بغدا خالصةال X)اذا كان " عرفها بالشكل اذ ،إن أول من قدم تعریف المجموعة شبه قبل المفتوحة هو الریاضي اندرجفك ,  ) فضاء تبولوجي وA مجموعة جزئیة منX فأنA ان تسمى مجموعة شبه قبل المفتوحة اذا ك � ⊆ � ∘��� . " أعالهلقد قمنا في هذا البحث بدراسة خواص المجموعات شبه قبل المفتوحة ولكن لیس عن طریق التعریف تعریف ثاني مكافئ لتعریفها وكذلك درسنا العالقة بینها وبین أنواع أخرى من المجامیع طة ابواسانما .المفتوحة الضعیفة IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Some Results on Semi-preopen Sets R. N. Majeed , A. I. Nasir Department of Mathematics,College of Education – Ibn Al- Haitham, University of Baghdad Abstract The definition of semi-preopen sets were first introduced by "Andrijevic" as were is defined by :Let (X ,  ) be a topological space, and let A ⊆ �, then A is called semi-preopen set if � ⊆ � ∘��� . In this paper, we study the properties of semi-preopen sets but by another definition which is equivalent to the first definition and we also study the relationships among it and (open, α-open, preopen and semi-p-open )sets. 1.Preliminaries Definition 1.1 (1)(4): A subset A of a topological space (X ,  ) is called α – open set if and only if A ح �°�����° The family of all α – open sets is denoted by α . Definition 1.2 (1)(5) : A subset A of a topological space ( X ,  ) is called a preopen set if A ح ��° The complement of a preopen set is called preclosed set . The family of all preopen sets of X is denoted by PO(X). The family of all preclosed sets of X is denoted by PC(X). Theorem 1.3 (2) : The union of any family of preopen sets is a preopen set. Definition 1.4 (1) : The intersection of all preclosed sets containing A is called the preclosure of A, denoted by pre-cl A. Definition 1.5 (1) : A subset A of a topological space ( X ,  ) is said to be semi-p-open set, if there exists a preopen set in X say U such that Uح A ح pre-cl U. The complement of a semi-p-open set is called semi-p-closed set. The family of all semi-p-open sets of X is denoted by S-P(X). The family of all semi- p-closed sets of X is denoted by S-PC(X). Proposition 1.6 (2): For any subset A of a topological space ( X,  ), pre-cl A ح A� and the converse is not true. IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 2. semi– preopen sets Definition 2.1(6 ): A subset A of a topological space ( X ,  ) is said to be semi-preopen set , if and only if there exists a preopen set in X say U such that U ح A ح �� The complement of semi-preopen set is called semi- preclosed set . The family of all semi-preopen sets of X is denoted by SPO(X). The family of all semi-preclosed sets of X is denoted by SPC(X). Definition 2.2: Let ( X ,  ) be a topological space and Aح X, A is called semi-preneighborhood of a point x in X, if there exists semi-preopen set U in X such that xخ U ح A . Theorem 2.3 : The union of any family of semi- preopen sets is semi- preopen set. Proof : Let { Aα }; α خL be any family of semi-preopen sets, we must prove ⋃ ���∈� is semi-preopen set. This means we must prove there exists UخPO(X) such that U ⋃ ح �∈� �� . �� ح Since for all αخL, Aα is semi- preopen , therefore there exists Uα خPO(X) such that �� ح �� ح ���, and since ⋃ ���∈� is a preopen set (by theorem 1.3 ), therefore let U = ⋃ ���∈� ,then we get U ح ⋃ ���∈� ………… (1) Now since �� ح ��� for all αخL, therefore ⋃ ���∈� ⋃ ح Lخ���� for all αخL, implies ⋃ Lخ� �� ⋃ ح Lخ��� ����������� for all αخL, thus ⋃ Lخ� �� �� ح = ⋃ Lخ��� ����������� ….. (2) and from (1) and (2) we get, there exists UخPO(X) such that U ح ⋃ �� �∈� ⋃ therefore ,�� ح �∈� �� is semi-preopen set. ■ Corollary 2.4 : The intersection of any family of semi-preclosed sets is semi-preclosed set . Proof: Let { Fα }; α خL be any family of semi-preclosed sets, we must prove ⋂ ���∈� is semi- preclosed this means we must prove (⋂ ���∈� ) � is semi-preopen set. Since for all αخL, �� � is semi-preopen set ( by Definition 2.1), therefore ⋃ �� � �∈� is semi-preopen set (by Theorem 2.3), implies there exists U PO(X) such that U ح ⋃ �� � �∈� ⋃ and since ,�� ح �� � �∈� = (⋂ ���∈� ) �, therefore U ح (⋂ ���∈� Thus .�� ح � ( (⋂ ���∈� ) � is semi-preopen set, implies ⋂ ���∈� is semi-preclosed set. ■ Remark 2.5 : The intersection of two semi-preopen sets need not to be semi-preopen set, as the following example shows: Example 1 : Let X={1,2,3},  ={X, , {1,2}} PO(X) =   {{1},{2},{1,3},{2,3}}, SPO(X) = PO(X) Let A= {1,3} and B={2,3} are both semi-preopen sets, but A  B={3} is not semi-preopen set. Remark 2.6 : The union of two semi-preclosed sets need not to be semi-preclosed set, as the example 1, Let X={1,2,3},  ={X, , {1,2}}then {1} and {2} are two semi-preclosed sets since X-{1}={2,3} and X-{2}={1,3} are two semi-preopen sets, but {1}{2}={1,2} is not semi-preclosed set since X- {1,2}={3} is not semi-preopen set. Definition 2.7 : The union of all semi-preopen sets contained in A is called the semi-preinterior of A, denoted by S-pre-int A . Definition 2.8 : The intersection of all semi-preclosed sets containing A is called the semi-preclosure of A, denoted by S-pre-cl A . Proposition 2.9: 1. If Aح B, then S-pre-int A ح S-pre-int B. .A . S-pre-int A 2 ح 3. S-pre-int A  S-pre-int B ح S-pre-int (A  B). 4. S-pre-int (A  B) ح S-pre-int A  S-pre-int B. Proof : The proof of (1) and (2) is direct by the definition of subsets and S-pre-int A . IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 3. Since Aح A, therefore S-pre-int Aح S-pre-int (A  B) (by part 1), and since ح A, therefore S-pre-int ح S-pre-int (A  B) (by part 1), implies S-pre-int A  S-pre- int B ح S-pre-int (A  B). The converse is not true in general, as the following of example (1): Let A={2}, ={3} and A={2,3}, then: S-pre-int{2}={2}, S-pre-int{3}=  and S-pre-int{2,3}={2,3}. ut S-pre-int (A  B) ={2,3} {2}= S-pre-int A  S-pre-int B. 4. A ح A, then this implies that S-pre-int (A  B) ح S-pre-int A (by part 1), and A S-pre-int  (by part 1), therefore S-pre-int (A  ح , then S-pre-int (A  B) ح B) ح S-pre-int A  S-pre-int B. But S-pre-int A  S-pre-int B  S-pre-int (A  B), as the following example shows: Example 2 : Let X={1,2,3,4}, ={X, ,{1},{2},{1,2}} PO(X) =   {{1,2,3},{1,2,4}} SPO(X) = PO(X)  {{1,3},{1,4},{1,3,4},{2,3},{2,4},{2,3,4}} let A ={2,3,4}, B ={1,3,4} and A  B= {3,4},then: S-pre-int {2,3,4}={2,3,4}, S-pre-int {1,3,4}= {1,3,4} and S-pre-int (A  B) =  .But S-pre-int A  S-pre-int B = {3,4}   = S-pre-int (A  B). Proposition 2.10 : 1. If Aح B, then S-pre-cl A ح S-pre-cl B. 2. A ح S-pre-cl A. 3. S-pre-cl  =  , S-pre-cl X = X. 4. S-pre-cl A  S-pre-cl B ح S-pre-cl (A  B). 5. S-pre-cl (A  B) ح S-pre-cl A  S-pre-cl B. Proof : 1. Let x be any element in X, such that x S-pre-cl B, which implies there existence of a semi-preclosed set F such that � ⊆ � and x F, but � ⊆ � so � ⊆ � and x F , hence x S- pre-cl A. 2. since S-pre-cl A is the intersection of all semi-preclosed sets containing A, we have A ح S- pre-cl A. 3.  and X are semi-preopen sets (by being open set), thus S-pre-cl  =  and S-pre-cl X = X. 4. Since A ح A B, therefore S-pre-cl A ح S-pre-cl (A  B) (by Part 1), and since B ح A B , therefore S-pre-cl B ح S-pre-cl (A  B) (by Part 1), implies S-pre-cl A  S-pre-cl B ح S-pre-cl (A  B). 5. Since A  B ح A, then this implies that S-pre-cl (A  B) ح S-pre-cl A (by Part 1), and since A  B ح B, therefore S-pre-cl (A  B) ح S-pre-cl B (by Part 1), therefore S-pre-cl (A  B) ح S-pre-cl A  S-pre-cl B. But the converse of part (4) is not true to see this, let A={1},B={2} and A B={1,2} in the example 2, then: SPC( X) ={X, ,{,3,4},{1,3,4},{3,4},{4},{3},{2,4},{2,3},{2},{1,4},{1,3},{1}} S-pre-cl {1}={1}, S-pre-cl {2}={2}and S-pre-cl {1} S-pre-cl {2}={1,2} but S-pre-cl ({1}{2}) = X, which shows S-pre-cl (A  B)  S-pre-cl A  S-pre-cl B And also, the converse of part (5) is not true to see this, Let A={1,2,3}, B={1,3,4} and A  B ={1,3} in the example 2, then: S PC( X) ={X, ,{,3,4},{1,3,4},{3,4},{4},{3},{2,4},{2,3},{2},{1,4},{1,3},{1}} S- pre- cl {1,2,3}=X, S-pre-cl {1,3,4}={1,3,4} and S-pre-cl {1,2,3} S-pre-cl{1,3,4}={1,3,4} But S-pre-cl ({1,2,3} {1,3,4}) ={1,3} which shows S-pre-cl A  S-pre-cl B  S-pre-cl (A  B). Proposition 2.11 : A is semi-preclosed set, if and only if A= S-pre-cl A. Proof: ( ) If A is semi-preclosed set, we must prove A= S-pre-cl A, that is mean we must prove A ح S-pre-cl A and S-pre-cl A ح A. Now, to prove S-pre-cl A ح A, since S-pre-cl A is the intersection of all semi- preclosed sets containing A, and since A is semi-preclosed set, and Aح A, implies S-pre-cl A ح A. Now, to prove A ح S-pre-cl A, let x be any element in X, such that x S-pre-cl A , which implies there existence of a semi-preclosed set F such that x∉ F, but A ح F, hence x∉ A. Thus A ح S-pre-cl A, which implies A= S-pre-cl A. (⟸) The prove is direct (by Corollary 2.4) . ■ Corollary 2.12 : S-pre-cl (S-pre-cl A) = S-pre-cl A . Proof: Since S-pre-cl A is semi-preclosed set (by Corollary 2.4), therefore S-pre-cl (S-pre-cl A) = S-pre-cl A (by Proposition 2.11). ■ Now, we give the connection between semi-preopen sets and some other kinds of weakly open sets. 3. Relationship among open, α-open, preopen, semi –p- open and semi-preopen sets Remark 3.1 (3) : 1. Every open set is a preopen set, but not conversely. 2. Every closed set is a preclosed set, but not conversely. Remark 3.2 : Every preopen set is semi-preopen set. Proof: Since A is a preopen set and Aح A, and since for any subset A of X, A ح A� , therefore there exists a preopen set A such that Aح A ح A� . Thus A is semi-preopen set. But the converse need not to be true in general, as the following of example 2 X={1,2,3,4}, ={X, ,{1},{2},{1,2}} PO(X) =   {{1,2,3},{1,2,4}} SPO(X) = PO(X)  {{1,3},{1,4},{1,3,4},{2,3},{2,4},{2,3,4}} it is clear that {1,3} is semi-preopen set, but it is not a preopen set. From remark 3.1 and remark 3.2 we obtain the following: Remark 3.3 : IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 Every open set is semi-preopen set . But the converse may be false, as the example 2 in remark 3.2 . Remark 3.4 : Every α-open set is semi-preopen set. Proof: Since A is α-open set, therefore A ح �°�����°, and since �° is open set this implies �° is a preopen set (by Remark 3.1). And �°ح A so �°ح A ح ح °�����°� �°��� , hence �°ح A ح �°����. Thus, A is semi-preopen set. ■ The converse of remark 3.4 is not true, as the following of example 2: X={1,2,3,4}, ={X, ,{1},{2},{1,2}} α = PO(X) =   {{1,2,3},{1,2,4}} SPO(X) = PO(X)  {{1,3},{1,4},{1,3,4},{2,3},{2,4},{2,3,4}}. Remark 3.5 : Every semi-p-open set is semi-preopen set. Proof: Let A be any semi-p-open set, this means there exists a preopen set in X say U such that U ح A ح pre-cl U, and since pre-cl U ح �� (by proposition 1.6 ), therefore U ح A ح ��. Thus A is semi-preopen set. ■ ut the converse is not true, as the following example show: Example 3 : Let X= {1,2,3,4 }, ={ , X, {1,2}, {3}, {1,2,3}}, �={ X,  , {1,2,4},{3,4},{4}} (X) =   {{1},{2},{1,3},{2,3},{1,3,4},{2,3,4}} C(X) = �  {{2,3,4},{1,3,4},{2,4},{1,4},{2},{1}} S(X) = (X)  {{1,4},{2,4},{3,4},{1,2,4}} now {1,4}خ S(X), but {1}{1,4} ح pre-cl {1}={1},thus {1,4} is not semi-p-open set. Conclusion 1. The union of any family of semi- preopen sets is semi- preopen set. 2. The intersection of any family of semi-preclosed sets is semi-preclosed set . 3. Every preopen set is semi-preopen set. 4. Every open set is semi-preopen set . 5. Every semi-p-open set is semi-preopen set. References 1. Navalagi,G.B. (2000), "Definition Bank in General Topology ", Internet. 2 . Esmaeel, R.B. (2004) " On Semi-P-Open Sets", M.Sc. thesis, University of Baghdad. 301, -4), 299-(356Math. Hungarica, Ganster and Ivan Rrilly, (1990), Acta Maximum 3. Internet. 4. Olav Njastad, (1965), pacific Journal of Mathematics, 15:3 . 5. Mshhour,A.S. Abd El-Monsef M.E. and El-Deeb, S.N. (1981). p roc.Math. and phys.Soc.Egypt 51. 6. Dontchev, J. (1994 ), Helsinki Unv., J. pure appl. Math., 25(9). 7. Navalagi, G.B. (2000). Definition Bank in General Topology, Department of Mathematics , G.H.College, karanataka,India