المجموعات فوق شبه المفتوحة من النوع α. On supra semi-α-open sets

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1 On supra semi-α-open sets وزارة التربية العراقية - المديرية العامة لتربية ذي قار. aasadalkhafaji@yahoo.com ملخص الملخص فييه اييلا القدييا مييوما موع ميي لايي م موعييو مييت المفم ايي ا المو يي فييه الو يي ال ق ل مه تواى المفم ا ف ق شقه المو مت الا م. α ي ا ا وي ن ريوبع أيو عري ام ميت اليووا الممي معا وم ومت خيل ايلا الموعي الوصل. الكلمات المفتاحية : المفم ا ا المو مت الاي م α,المفم اي ا شيقه المو ي ميت الايي م α,المفم ايي ا فيي ق المو يي مييت الايي م α, الو يي اا فيي ق ال ق ل مايي, الواليي semi-α-t 1 و الو يي المميي معا مييت الايي م α, الو يي,semi-α-T 0 الو يي.semi-α- T 2 3 مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

2 Abstract On supra semi-α-open sets ASAAD SHAKIR HAMEED AL-KHAFAJI Ministry OF Education, Iraq In this paper we introduce a concept fora new type ofopen setsintopologicalspacecalledthe supra semi-α-open set. Through this conceptwewillexaminesome typesofcontinuousfunctionsandseparation axioms. Key words : α-open sets, semi-α-open, supra-α-open sets, supra topological spaces,α-continuous functions, semi-α-t 0 space, semi-α- T 1 space and semi-α-t 2 space. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

3 3.3.3 Example : If we look at theexample 3.3.2we find that the supra semi-α-t 2 spacebut it is not T 2 space, because 1 2There are no two setsareopen, such as T,G so, onecontainingtheelement1and the othercontainingthe element2and T G =. References : [1] O.Njastad,1965-on some classes of nearly open sets.pacific,j.math.15, [2] A.S.Mashhour,A.A Allam, F.S.Mahmoud and F.H.Khedr,1983-on supra topological spaces, Indian J.Pure and Appl.Math.no.4, 14, [3] I.Reilly and M.vamanamurthy, 1985-On - continuity in topological spaces, Acta Math.Hungar., 45, [4]G.B. Navalagi Definition Bank in General Topology [5] R. Devi,S.Sampathkumar and M.Calads,2008-on supra--open sets and S-continuous function,general Mathematics,vol.16,Nr.2, مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

4 3.3.1 Remark: Every T 0 space is supra semi-α-t0 space. The opposite situation is not necessarily true Example : Let ={1,2,3},,{3}}Then,{3},{1,3},{2,3}} space but not T 0 We note that is supra semi-α-t 0 no opensetso thatcontainonewithouttheother. because 12There are Remark : Every T space is supra semi-α- 1 T space. 1 The opposite situation is not necessarily true Example : Let ={1,2,3},,{1},{2},{1,2},{2,3}},{1},{2},{1,2}} Then space but not T 1 We note that is supra semi-α-t 1 because 23There are no two setsareopenso thatoneofthemcontainingelement 2andtheothercontainingtheelement Remark : Every T space is supra semi-α- 2 T space. 2 The opposite situation is not necessarily true. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

5 Let be supra semi--t0 space, x y in Hence there exists G an supra semi-- open set in such that x G, y G or x G, y G Then G c is supra semi--closed set and x G c, y G c Therefore x ss cl {y} (since x G c Hence ss cl {x} ss cl {y} Proposition: A space is supra semi--t1 space if and only if {x} is supra semi--closed set, x. Proof: Let be supra semi--t1 space. Let p, to prove {p} is supra semi--closed set. {p} c \{p} x p in Hence there exists an supra semi--open set G such that x G, p G or x G, p G. If xg, pg x G {p} c {p} c is an supra semi--open set {p} is supra semi--closed set. Let {p} be an supra semi--closed set, p, to prove is supra semi--t1 space. Let x y in Hence {x}, {y} are supra semi--closed sets {x} c, {y} c are Supra semi-- open sets and y {x} c, x {x} c, x {y} c, y {y} c Therefore is supra semi--t1 space. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

6 Supra semi-α-t 1 space Definition : A space is said to be supra semi-α-t space if for each pair of distinct points 2 x, y in there exist two supra semi-α - open sets T and G such that x T, y G and T G =.. x T. y G Supra semi-α-t 2 space Theorem: A space is a supra semi-α-t space if and only if ss cl {x} ss cl {y} for each x y in. Proof: Let ss cl {x} ss cl {y} for each x y in Hence ss cl {x} ss cl {y} or ss cl {y} ss cl {x} Suppose that ss cl {x} ss cl {y} x ss cl {y} x 0 ( sscl { y} c but c ( sscl { y} is an supra semi--open set and y ( sscl { y} c Therefore is -T0 space. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

7 3.3.1 Definition: A space is said to be supra semi-α-t0 space if for each pair of distinct points x, y in there exist G supra semi-α -open set of containing one point but not the other.. x G. y Supra semi-α-t 0 space Definition : A space is said to be supra semi-α-t space if for each pair of distinct points 1 x, y in there exist two supra semi-α-open sets U and V containing x and y respectively, such that y U, x V. x U V. y مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

8 3.2.2 Theorem: Every supraα-continuous function is a supra semi-α-continuous function. Proof: Let f : Y be a supraα-continuous function.therefore f 1 ( A is a supra α- open set in for each open set A in Y. Since : Every supra α-open set is supra semi-α-open set, This implies f 1 ( A supra semi-α-open set in. is Hence f is a supra semi-α-continuous function. The converse of the above theorem need not be true. This is show by the following example Example : Let ={a,b,c},,{ a},{ b},{ a, b}} Then Let f : where : f ( a a, f ( b f ( c b supra semi-α-continuous function but not supra α-continuous f because {b} is open set in and f 1 ({ b} { b, c} But { b, c} is not supra α-open set in. 3.3 Supra semi-α-t i spaces, i 0,1,2 In this section we introduce a new class of separation axioms. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

9 3.2.2 Definition: Let (, and ( Y, be two topological spaces and supra topology with and respectively. we define a function f, are associated : Y to be a supra semi-α-continuous function if the inverse image of each open set in Y is supra semi- α-open set in (, [5] 3.2.1Theorem: Every continuous function is a supra α-continuous function. The converse of the above theorem need not be true. This is show by the following example Example: Let ={a,b,c,d},,{ a}},{ a},{ a, b},{ a, c},{ a, d},{ a, b, c},{ a, b, d},{ a, c, d}} Let Y={x,y,z}, Y Y,{ x}} Y Y,{ x},{ x, y},{ x, z}} Let f : Y where: f ( a f ( b x, f ( c y, f ( d z supraα-continuous function but not continuous function because f {x} is open set in Y and f 1 ({ x} { a, b} But { a, b} is not open set in. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

10 The supra semi-α-interior of a set A is denoted byss int ( A int ( A = {B:B is a supra semi- α-open set and A B and defined as :ss } Theorem: Any union of supra semi-α-open set is always a supra semi- α-open set. Proof: Let { A i } ij family of supra semi-- α-open set, this is Supra α-open set such that : G i A i supra cl( A i ; i J Gi supra cl ( G i A i A i ( supra cl ( A i supra cl ( A i Gi supra semi- α-open set. { A i } ij Corollary : Any intersection of supra semi- α-closed set is always a supra semi- α- closed set. 3.2 Supra semi-α-continuous function In this section we introduce a new class of functions [5] Definition: Let (, and ( Y, be two topological spaces and associated supra topology with and respectively. we define a function f, are : Y to be a supra α-continuous function if the inverse image of each open set in Y is supra α-open set in (,. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

11 3.1.1 Remark: Complement of a supra semi- α-open set is called a supra semi-αclosed set. Every supra open set is supra α-open set The following example shows that the converse of the above remark is not true Example : Remark: Let (, be a supra topological space where ={a,b,c} and = {,,{a}}. Here,{a,b} is a supra α-open set but not supra open set. Every supra α-open set is supra semi-α-open set Example : The following example shows that the converse of the above remark is not true. Supra open set Let (, be a supra topological space.where ={a,b,c,d} and = {,,{a},{b},{a,b},{a,b,c}}. Here, {b,c} is a supra semi-α-open set, but not a supra α-open set. From the remarks above we get the following : supraα-open set supra semi-α-open set Definition : The supra semi-α- closure of a set A is denoted by ss cl (A (A cl = {B:B is a supra semi- α-closed set and A B } and defined as :ss مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

12 [2] 2.9 Definition: Let (, be a topological space and a supra topology associated with if. be a supra topology on. we call [2] 2.10 Definition: Let (, and ( Y, two be a topological spaces, Let supra topologies with and respectively. Let f : Y be a map from into Y, then f and are associated is a supra continuous function if the inverse image of each open set in is supra open set in 2.11 Definition: Let (, f : Y be a function, then f is said to be Semi-α-continuous function if and only if for each A open set in Y, then f 1 ( A is a semi-α-open set in. 3- Results and discussions : 3.1 Basicproperties of supra semi-α-open sets In this section we introduce a new class of sets Definition:. Let (, be a supra topological space. A set A is called : (i supra α-open set if : A supra int(supra cl (supra int(a. Complement of a supra α-open set is called a supra α-closed set. (ii supra semi-α-open set if: U A supra cl(a, and U is supra α-open set in (, مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

13 [4] 2.4 Definition: Let (, be topological space.a set A is called semi-α-open set if U A cl(u and U isα-open set in (,. Complement of a semi-α-open set is called a semi-α-closed set. [4] 2.5 Definition : A space is said to be semi-α-t 0 space if for each pair of distinct points in there exist semi-α-open set of containing one point but not the other. [4] 2.6 Definition: A space is said to besemi-α-t space if for each pair of distinct points 1 x, y in there exist a two semi-α-open sets U and V containing x and y respectively, such that y U, x V. [4] 2.7 Definition : A space is said to be semi-α-t space if for each pair of distinct points 2 there exist two a semi-α-open sets G and 1 G such that 2 x G, y 1 G 2 and G 1 G 2 =. [2] 2.8 Definition : The supra closure of a set A is denoted by scl(a and defined as: x, y in scl(a = {B:B is a supra closed set and A B }. The supra interior of a set A is denoted by sint(a and defined as: sint(a = {B:B is a supra open set and B A }. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول

14 1-Introduction: In 1965, Najastad [1] introduced the α-open sets. In 1983, A.S.Mashhour [2] introduced the supra topological spaces and studied supra continuous function. In 1985, I.Reilly and M.vamanamurthy[3]introduced α-continuous functions. In 2000,G.B.Navalagi[4] introduced the semi-α-open sets. In 2008,R. Devi,S.Sampathkumar and M.Calads[5] introduced the supra α- open sets and Sα-continuous function. Now, we introduce the concept of supra semi-α-open sets and supra semi-αcontinuous and investigate some of the basic properties for this class of functions and study some types of separation axioms in topological spaces. 2. Definitions and concepts : 2.1[1] Definition: Let (, be topological space.a set A is called α-open set if : A int (cl(int(a. Complement of a α-open set is called a α-closed set. [2] 2.2 Definition: A subfamily of is said to be a supra topology on if: (1,. (2 if A i for all i J, then A i. is called a supra topological space. The elements of (, are called supra open sets in supra open sets is called a supra closed set. [3] 2.3 Definition: (, and complement of Let (, and ( Y, be two topological spaces.a function f : Y is called α-continuous function if the invers image of each open set in Y is α-open set in. مجلة أبحاث البيئة والتنمية المستدامة العدد الثاني المجلد االول