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 Ais 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-P C(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 . 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, {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 : The proof of (1) and (2) is clear by the definition of subsets and 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. The proof is similarly to the proof of part (3) in proposition 2.9 . 5. The proof is similarly to the proof of part (4) in proposition 2.9 . 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: Necessity , clear. Sufficiency. The proof is direct (by corollary 2.4). Corollary 2.12 : S-pre-cl (S-pre-cl A) = S-pre-cl A . IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 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 : 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. IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 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, 990), Acta Ganster and Ivan Rrilly, (1 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