2) 3( 22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد 009 قبل المفتوحة شبه المجموعاتبعض النتائج حول احمد إبراهیم ناصررشا ناصر مجید ، جامعة بغداد الهیثم ،أبن -كلیة التربیة سم الریاضیات،ق 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 Departme nt of Mathematics,College of Education – Ibn Al- Haitham, Unive rsity of Baghdad Abstract The definition of semi-preopen sets were first introduced by "Andrijevic" as were is defined by :Let (X ,  ) be a top ological sp ace, and let A ⊆ �, then A is called semi-p reopen set if � ⊆ � ∘��� . In this p aper, we study the prop erties of semi-preopen sets but by another definition which is equivalent t o the first definition and we also st udy the relationships among it and (op en, α-op en, p reop en and semi-p -op en )sets. 1.Preliminaries De finition 1.1 (1)(4): A subset A of a top ological sp ace (X ,  ) is called α – open set if and only if A ح �°����� ° The family of all α – op en sets is denoted by α . De finition 1.2 (1)(5) : A subset A of a top ological sp ace ( X ,  ) is called a p reopen 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). The orem 1.3 (2) : The union of any family of p reop en sets is a p reop en set. De finition 1.4 (1) : The intersection of all p reclosed sets containing A is c alled the preclosure of A, denoted by p re-cl A. De finition 1.5 (1) : A subset A of a top ological sp ace ( X ,  ) is said to be semi-p-open set, if there exist s a p reop en set in X say U such that Uح A ح p re-cl U. The complement of a semi-p-op en set is called semi-p -closed set. The family of all semi-p -op en 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). Proposi tion 1.6 (2): For any subset A of a top ological sp ace ( X,  ), p re-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 De finition 2.1(6 ): A subset A of a top ological sp ace ( X ,  ) is said to be semi-preopen set , if and only if there exist s a p reop en set in X say U such that U ح A ح �� The complement of semi-p reopen set is called semi- p reclosed set . The family of all semi-p reopen sets of X is denoted by SPO(X). The family of all semi-p reclosed sets of X is denoted by SPC(X). De finition 2.2: Let ( X ,  ) be a top ological sp ace 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 . The orem 2.3 : The union of any family of semi- p reopen sets is semi- preop en set. Proof : Let { Aα }; α خL be any family of semi-p reopen sets, we must p rove ⋃ ���∈� is semi-p reopen set. This means we must p rove there exists UخPO(X) such that U ⋃ ح �∈� . �� ح �� Since for all αخL, Aα is semi- p reopen , 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 ⋃ L, thusخL����������� for all αخ��� Lخ� �� ⋃ = �� ح L����������� ….. (2) andخ��� from (1) and (2) we get, t here exists UخPO(X) such that U ح ⋃ �� �∈� ⋃ therefore ,�� ح �∈� �� is semi-p reopen set. ■ Corollary 2.4 : The intersection of any family of semi-p reclosed sets is semi-p reclosed set . Proof: Let { Fα }; α خL be any family of semi-p reclosed sets, we must p rove ⋂ ���∈� is semi- p reclosed this means we must p rove (⋂ ���∈� ) � is semi-p reopen set. Since for all αخL, �� � is semi-p reopen set ( by Definition 2.1), therefore ⋃ �� � �∈� is semi-p reopen 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-p reopen set, imp lies ⋂ ���∈� is semi-p reclosed set. ■ Remark 2.5 : The intersection of two semi-p reopen sets need not to be semi-preopen set, as the following example shows: Exam ple 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-p reopen sets, but A  B={3} is not semi-p reopen set. Remark 2.6 : The union of two semi-preclosed sets need not to be semi-preclosed set, as t he examp le 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-p reop en sets, but {1}{2}={1,2} is not semi-p reclosed set since X- {1,2}={3} is not semi-p reop en set. De finition 2.7 : The union of all semi-p reopen sets contained in A is called the semi-preinterior of A, d enoted by S-p re-int A . De finition 2.8 : The intersection of all semi-p reclosed sets containin g A is called the semi-preclosure of A, denoted by S-p re-cl A . Proposi tion 2.9: 1. If Aح B, then S-p re-int A ح S-p re-int B. A . S-p ح re-int A 2. 3. S-p re-int A  S-p re-int B ح S-p re-int (A  B). 4. S-p re-int (A  B) ح S-p re-int A  S-p re-int B. Proof : The proof of (1) and (2) is direct by the definition of subsets and S-p re-int A . IBN AL- HAITHAM J. FO R PURE & APPL. SC I VO L.22 (3) 2009 3. Since Aح A, therefore S-p re-int Aح S-p re-int (A  B) (by p art 1), and since ح A, therefore S-p re-int ح S-p re-int (A  B) (by p art 1), imp lies S-p re-int A  S-p re- int B ح S-p re-int (A  B). The converse is not true in gener al, as the following of examp le (1): Let A={2}, ={3} and A={2,3}, then: S-p re-int{2}={2}, S-pre-int{3}=  and S-p re-int{2,3}={2,3}. ut S-p re-int (A  B) ={2,3} {2}= S-p re-int A  S-p re-int B. 4. A ح A, then this implies that S-pre-int (A  B) ح S-p re-int A (by p art 1), and A , then S-p ح re-int (A  B) ح S-p re-int  (by p art 1), therefore S-p re-int (A  B) ح S-p re-int A  S-p re-int B. But S-p re-int A  S-p re-int B  S-p re-int (A  B), as the following examp le shows: Exam ple 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-p re-int {2,3,4}={2,3,4}, S-pre-int {1,3,4}= {1,3,4} and S-p re-int (A  B) =  .But S-p re-int A  S-p re-int B = {3,4}   = S-p re-int (A  B). Proposi tion 2.10 : 1. If Aح B, then S-p re-cl A ح S-p re-cl B. 2. A ح S-p re-cl A. 3. S-p re-cl  =  , S-p re-cl X = X. 4. S-p re-cl A  S-p re-cl B ح S-p re-cl (A  B). 5. S-p re-cl (A  B) ح S-p re-cl A  S-p re-cl B. Proof : 1. Let x be any element in X, such that x S-p re-cl B, which imp lies there existence of a semi-p reclosed set F such that � ⊆ � and x F, but � ⊆ � so � ⊆ � and x F , hence x S- p re-cl A. 2. since S-p re-cl A is the intersection of all semi-preclosed sets containin g A, we have A ح S- p re-cl A. 3.  and X are semi-p reopen sets (by being op en set), thus S-pre-cl  =  and S-p re-cl X = X. 4. Since A ح A B, therefore S-p re-cl A ح S-p re-cl (A  B) (by Part 1), and since B ح A B , therefore S-p re-cl B ح S-p re-cl (A  B) (by Part 1), imp lies S-p re-cl A  S-p re-cl B ح S-p re-cl (A  B). 5. Since A  B ح A, t hen this imp lies that S-pre-cl (A  B) ح S-p re-cl A (by Part 1), and since A  B ح B, therefore S-p re-cl (A  B) ح S-p re-cl B (by Part 1), therefore S-p re-cl (A  B) ح S-p re-cl A  S-p re-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-p re-cl {1}={1}, S-pre-cl {2}={2}and S-p re-cl {1} S-p re-cl {2}={1,2} but S-p re-cl ({1}{2}) = X, which shows S-p re-cl (A  B)  S-p re-cl A  S-p re-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- p re- cl {1,2,3}=X, S-pre-cl {1,3,4}={1,3,4} and S-p re-cl {1,2,3} S-p re-cl{1,3,4}={1,3,4} But S-p re-cl ({1,2,3} {1,3,4}) ={1,3} which shows S-p re-cl A  S-p re-cl B  S-p re-cl (A  B). Proposi tion 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 p rove A= S-p re-cl A, that is mean we must p rove A ح S-p re-cl A and S-p re-cl A ح A. Now, to p rove S-p re-cl A ح A, since S-p re-cl A is the intersection of all semi- p reclosed sets containing A, and since A is semi-p reclosed set, and Aح A, implies S-p re-cl A ح A. Now, to p rove A ح S-p re-cl A, let x be any element in X, such that x S-p re-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-p re-cl A, which implies A= S-p re-cl A. (⟸) The prove is direct (by Corollary 2.4) . ■ Corollary 2.12 : S-p re-cl (S-p re-cl A) = S-pre-cl A . Proof: Since S-p re-cl A is semi-preclosed set (by Corollary 2.4), therefore S-p re-cl (S-p re-cl A) = S-p re-cl A (by Prop osition 2.11). ■ Now, we give the connection between semi-p reopen sets and some other kinds of weakly op en sets. 3. Relationship among open, α-open, preopen, semi –p- open and semi-preopen sets Remark 3.1 (3) : 1. Every op en set is a preop en set, but not conversely. 2. Every closed set is a preclosed set, but not conversely. Remark 3.2 : Every p reop en set is semi-p reopen set. Proof: Since A is a preopen set and Aح A, and since for any subset A of X, A ح A� , therefore there exist s a p reop en set A such that Aح A ح A� . Thus A is semi-p reop en set. But the converse need not t o be true in general, as t he following of examp le 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-preop en set, but it is not a preop en set. From remark 3.1 and r emark 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 op en set is semi-p reop en set . But the converse may be false, as t he examp le 2 in remark 3.2 . Remark 3.4 : Every α-open set is semi-p reopen set. Proof: Since A is α-op en set, t herefore A ح �°����� ° , and since �° is op en set this imp lies �° is a preopen set (by Remark 3.1). And �°ح A so �°ح A ح �°����� ° .����°� ح A ح°� hence , ���°� ح Thus, A is semi-p reopen set. ■ The converse of remark 3.4 is not true, as t he 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 -op en set is semi-preop en set. Proof: Let A be any semi-p -op en set, this means there exists a p reopen set in X say U such that U ح A ح p re-cl U, and since pre-cl U ح �� (by p rop osition 1.6 ), therefore U ح A ح ��. Thus A is semi-p reopen set. ■ ut the converse is not true, as t he following examp le show: Exam ple 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  p{1,4} ح{1} re-cl {1}={1},thus {1,4} is not semi-p -op en set. Conclusion 1. The union of any family of semi- p reopen sets is semi- preop en set. 2. The intersection of any family of semi-p reclosed sets is semi-p reclosed set . 3. Every p reop en set is semi-p reopen set. 4. Every op en set is semi-p reop en set . 5. Every semi-p -op en set is semi-preop en set. Re ferences 1. Navalagi,G.B. (2000), "Definition Bank in General Top ology ", Internet. 2 . Esmaeel, R.B. (2004) " On Semi-P-Op en Sets", M .Sc. thesis, University of Baghd ad. 301, -4), 299-(356M ath. Hungarica, Ganst er and Ivan Rrilly, (1990), Acta M aximum 3. Internet. 4. Olav Njast ad, (1965), pacific Journal of M athematics, 15:3 . 5. Mshhour,A.S. Abd El-M onsef M .E. and El-Deeb, S.N. (1981). p roc.M ath. and p hy s.Soc.Egy pt 51. 6. Dont chev, J. (1994 ), Helsinki Unv., J. p ure app l. M ath., 25(9). 7. Navalagi, G.B. (2000). Definition Bank in General Top ology , Dep artment of M athematics , G.H.College, karan ataka,India