Generalized (ST, PT)-Supra Closed Set In Bi-Supra Topological Space

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1 AENSI Journals Australian Journal of Basic and Applied Sciences ISSN: Journal home page: Generalized (ST, PT)-Supra Closed Set In Bi-Supra Topological Space 1 Tahh H. Jasim and 2 Firas N. Jasim 1 Depa.of Mathematics/college of computer Science and Mathematics 2 Depa.of Mathematics/college of Education Tikrit university A R T I C L E I N F O Article history: Received 25 June 2014 Received in revised form 8 July 2014 Accepted 25 July 2014 Available online 20 August 2014 A B S T R A C T This paper introduces generalized of (ST, PT)-supra closed set in bi-supra topological space and investigate the relations among some properties of them. Keywords: 2014 AENSI Publisher All rights reserved. To Cite This Article: Tahh H. Jasim and Firas N. Jasim., Generalized (ST,PT)-Supra Closed Set In Bi-Supra Topological Space. Aust. J. Basic & Appl. Sci., 8(13): , 2014 INTRODUCTION About thirty nine years ago, Kelly (1963) introduced the concept of bi- topological space where a set X equipped with two topologies and denoted by (X,τ 1,τ 2 ) where τ 1,τ 2 are two topologies defined on X. Al mashhour in (1983) introduced the concept of supra topological space as A subfamily τ of a famly of all subset of X is said to be a supra topology on X if a)x, φ τ b)if Ai τ for all i I then Aiϵτ, where I is index set. (X,τ) is called a supra topological space. The elements of τ are called supra open sets in (X,τ) and the complement of supra open set is called a supra closed set. In this paper we Introduce a Generalized (ST, PT)- closed set in bi-supra topological spaces. 2- Preliminaries: Let us recall the definitions and results which are used in the sequel. Definition 2.1: A subset A of a topological space (X,τ) is called. a) a pre-open set (Mashhour, S., M.E. Abd El-Monsef, 1983) if A int(cl(a)) and a pre-closed set if cl(int(a)) A; b) a semi-open set (Levine, N., 1963) if A cl(int(a)) and a semi-closed set if int(cl(a)) A; c) an α-open set (Njastad, O., 1965) if A int(cl(int(a))) and an α-closed set if cl(int(cl(a))) A; d) an β-open set (Andrijevic, D. 1986) if A cl(int(cl(a))) and an β-closed set if int(cl(int(a))) A; Remark 2.2: (Kelley, J.C., 1963)the set of all semi-open [resp. pre-open, β-open ] is a supra topology on X and the set of all α-open is α topology on X. Definition 2.3: (Maki, H., R. Devi, 1994) A subset A of a topological space (X,τ) is called., a) Scl(A) = { F: A F, F is s-closed set } b) Pcl(A)= { F: A F, F is p-closed set } c) αcl(a)= { F: A F, F is α-closed set } d) βcl(a)= { F: A F, F is β-closed set } Corresponding Author: Firas N. Jasim, Depa.of Mathematics/college of Education Tikrit university. ferisnaje@yahoo.com

2 240 Tahh H. Jasim and Firas N. Jasim, 2014 Definition2.4: A subset A of a topological space (X,τ) is called. 1) g-closed (Fukutake, T., 1986) if cl(a) U whenever A U and U is open set in X. 2) sg-closed (Fathi, H., Khedr, 2009) if-scl(a) U whenever A U and U is semi open set in X. 3) gs-closed (EL, O.A., Tantawy, 2005) if scl(a) U whenever A U and U is open set in X. 4) pg-closed (EL, O.A., Tantawy, 2005) if-pcl(a) U whenever A U and U is pre open set in X. 5) gp-closed (EL, O.A., Tantawy, 2005) if -pcl(a) U whenever A U and U is open set in X. 6) αg-closed [11] if-αcl(a) U whenever A U and U is open set in X. 7) gα-closed (EL, O.A., Tantawy, 2005) if -αcl(a) U whenever A U and U is α-open set in X. 8) gβ-closed (EL, O.A., Tantawy, 2005) if βcl(a) U whenever A U and U is open set in X. 9) w-closed (Fukutake, T., P. Sundharam, 2002) if cl(a) U whenever A U and U is semi open in X. The complements of the above sets are called their respective open sets. 3- bi-supra topological spaces: Now we introduce a new concept in bi-supra topological space and investigate some characterizent of it s: Definition 3.1: Let X be a non-empty set. Let ST be the set of all semi open subset of X(for short So x [7] and Let PT be the set of all α-open subset of X (for short Po(x))[8], then we say that (X,ST, PT) is a bi-supra topological space. when each of (X,ST ) and (X, PT) are supra topological space. Remark 3.2: the deference between bi-topological space (Maki, H., R. Devi, 1994) and bi-supra topological space. In this case the definition is (X,ST, PT) generalized to bi-topological space Kelly (Levine, N., 1963). Remark 3.3: It is clear that ST, PT was independent. Example 3.4: Let X={1,2,3,4} with τ={φ,{3},{1,2},{1,2,3},x} there for So x = ST ={Φ,{3},{1,2}, {1,2,3},{3,4},{1,2,4},X} Po x = PT={Φ,{3},{1,2},{1,2,3},{1},{2},{1,3},{2,3},{1,3,4},{2,3,4},X}. Hence (X,ST, PT) is bi-supra topological space. Now we introduce the definition of the type of open sets in bi-supra topological space. Definition 3.5: Let (X,ST, PT) be a bi-supra topological space. and Let G be a subset of X. Then G is said to be 1. (ST, PT)-supra open set if G=AUB where A Sτ and B Pτ.The complement (ST, PT)-supra open set is called (ST, PT)-supra closed set. 2. (ST, PT)*-supra open set if G=AUB where A Sτ, B Pτ such that B Sτ or A PT, B ST such that B PT. The complement of (ST, PT)*-supra open set is called (ST, PT)*-supra closed set. 3. bi-supra open set if G=A where A τ and A ατ. The complement of bi-supra open set is called bi-supra closed set. Proposition 3.6: 1. Every bi-supra open set is (ST, PT)-supra open set and every bi-supra closed set is (ST, PT)-closed set but the converse is not true. 2. Every (ST, PT)*-supra open set is (ST, PT)-supra open set and every (ST, PT)*-supra closed set is (ST, PT)-supra closed set but the converse is not true. 3. The (ST, PT)*-supra open set, bi-supra open set are independent and the (ST, PT)*-supra closed set, bisupra closed set are independent. Proof. Directly from definition. Remarks 3.7: Observe that The set of all (ST, PT)[ rep. (ST, PT)*, bi]-supra open sets and (ST, PT)[ rep. (ST, PT)*, bi]-supra closed set is need not necessarily form a topology it is a supra topology. Now we give some examples to explain the types of open sets in bi-supra topological space. Example 3.8:

3 241 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4} τ={φ,{1 { {2},{1,2},{1,2,3},{1,2,4},X} Sτ={φ,{1},{2},{1,2},{1,2,3},{1,2,4},{1,3},{1,4},{2,3}{2,4},{1,3,4},{2,3,4},X}PT ={φ,{1 { {2},{1,2},{1,2,3},{1,2,4},X} (ST, PT)-supra open set={φ,{1},{2},{1,2},{1,2.3},{1,2.4},{1,3},{1,4},{2,3},{2,4},{1,3,4},{2,3,4},x} (ST, PT)-supra closed set={x,{2,3,4},{1,3,4},{3,4},{4},{3}{2,4},{2,3},{1,4},{1,3},{2},{1},φ} (ST, PT)*-supra open set={φ,{1,3},{1,4},{2,3},{2,4},{1,3,4},{2,3,4},{1,2,3},{1,2,4},x} (ST, PT)*-supra closed set={x,{2,4},{2,3},{1,4},{1,3},{2},{1},{4},{3},x} bi-supra open set ={φ,{1}{2},{1,2},{1,2,3},{1,2,4},x} bi-supra closed set={x,{2,3,4},{1,3,4},{3,4},{4},{3},x} Remarks 3.9: Observe that (ST, PT)-supra open set is equivalence to b-supra open set. Although b-supra open set in supra topological space and (ST, PT)-supra open set in bi-supra topological space. The important point in the definition (3.5) that is (ST, PT)-supra open set is called bi-supra topology. 4- Generalized (ST, PT)- closed set in bi-supra topological space: In this section we introduce some new concept of generalized (ST, PT)-closed set in(x, ST, PT). Definition 4.1: A subset A of bi-supra topological space (X, ST, PT) is called. 1. ( ST, PT)*cl(A)= {F : A F, F is (ST, PT)*-supra closed} 2. bi-cl(a)= {F : A F: A F, F is bi-supra closed} 3. s(st, PT)*cl(A)= { F: A F, F is s(st, PT)*-supra closed} 4. s-bi-cl(a)= {F : A F, F is s-bi-supra closed } 5. p(st, PT)*cl(A)= {F: A F, F is p(st, PT)*-supra closed} 6. p-bi-cl(a)= {F:A F, F is p-bi-supra closed} 7.β(ST, PT)*cl(A)= {F:A F, F is β(st, PT)*-closed} 8. β-bi-cl(a)= {F: A F, F is β-bi-supra closed} 9. α(st, PT)*cl(A)= {F: A F, F is α(st, PT)*-supra closed} 10. α-bi-cl(a)= {F:A F, F is α-bi-supra closed} Definition 4.2: A subset A of bi-supra topological space (X, ST, PT) is called 1. (ST, PT)* int(a)=u{g:g A, G is (ST, PT)*-supra open} 2. bi-int(a)=u{g:g A, G is bi-supra open} 3. s(st, PT)*int(A)=U{G:G A. G is s(st, PT)*-supra open} 4. s-bi-int(a)=u{g:g A, G is s-bi-supra open} 5. p(st, PT)*int(A)=U{G:G A, G is p(st, PT)*-open} 6. p-bi-int(a)=u{g:g A, G is p-bi-supra open} 7. β(st, PT)*int(A)=U{G:G A, G is β(st, PT)*-supra open} 8. β-bi-int(a)=u{g:g A, G is β-bi-supra open } 9. α(st, PT)*int(A)=U{G:G A, G is α(st, PT)*-supra open} 10. α-bi-int(a)=u{g:g A, G is α-bi-supra open} Definition 4.3: A subset A of a bi-supra topological space (X, ST, PT) is called 1. semi(st, PT)*-supra open if A (ST, PT)*-cl((ST, PT)*-int(A)). 2. semi-bi-supra open if A bi-cl(bi-int(a)). 3. pre(st, PT)*-supra open if A (ST, PT)*-int((ST, PT)*-cl(A)). 4. pre-bi-supra open if A bi-int(bi-cl(a)). 5. β(st, PT)*-supra open if A (ST, PT)*-cl((ST, PT)*-int((ST, PT)*-cl(A))). 6. β-bi-supra open if A bi-cl(bi-int(bi-cl(a))). 7. α(st, PT)*-supra open if A (ST, PT)*-int((ST, PT)*-cl((ST, PT)*-int(A))).

4 242 Tahh H. Jasim and Firas N. Jasim, α-bi-supra open if A bi-int(bi-cl(bi-int(a))). The complements of the above mentioned open sets are called closed set (respectively). Definition 4.4: A subset A of a bi-supra topological spaces (X, ST, PT) is called. 1. g-( ST, PT)*[resp. gs-( ST, PT)*, gβ-( ST, PT)*, gp-( ST, PT)*, αg-( ST, PT)*]-closed if (ST, PT)*cl(A) U[resp. s(st, PT)*cl(A) U, β(st, PT)*cl(A) U, p(st, PT)*cl(A) U, α(st, PT)*cl(A) U ] whenever A U, U is (ST, PT)*-open set in X. 2. sg-( ST, PT)*[resp. w-( ST, PT)*, gα-( ST, PT)*, pg-( ST, PT)*]-closed if s(st, PT)*cl(A) U[resp. (ST, PT)*cl(A) U, α(st, PT)*cl(A) U, p(st, PT)*cl(A) U] whenever A U, U is s(st, PT)*[resp. s(st, PT)*, α(st, PT)*, p(st, PT)*]-open set in X. 3. g-bi[resp. gs-bi, gβ-bi, gp-bi, αg-bi]-closed if bi-cl (A) U[resp. s-bi-cl(a) U, β-bi-cl(a) U, p-bicl(a) U, α-bi-cl(a) U] whenever A U, U is bi-open set in X. 4. sg-bi[resp. w-bi, gα-bi, pg-bi ]-closed if s-bi-cl(a) U[resp. bi-cl(a) U, α-bi-cl(a) U, p-bicl(a) U] whenever A U, U is s-bi[resp. s-bi, α-bi, p-bi ]-open set. The complements of the above mentioned open sets are called closed sets(respectively). Proposition 4.5: Every (ST, PT)*[bi] supra closed is g-( ST, PT)*[g-bi] supra closed but not conversely. Let A X be a (ST, PT)*[bi] closed and Let A U where U is (ST, PT)*[bi]-open set since A is a (ST, PT)*[bi]-closed then(st, PT)[bi]-cl(A) =A U. Hence A is g-( ST, PT)*[g-bi]-closed Example 4.6: Let X={1,2,3,4} with τ={φ,{3},{1,2},{3,4}{1,2,3},x}.the set {3}is g-( ST, PT)*[g-bi]-supra closed but not (ST, PT)*[g-bi]-supra closed. Proposition 4.7: Every g-( ST, PT)*[g-bi]-supra closed is gβ-( ST, PT)*[gβ-bi]-supra closed but not conversely. Let A X be a g-( ST, PT)*[g-bi]-supra closed and Let A U where U is(st, PT)*[bi]-supra open set. since A is g-( ST, PT)*[g-bi]-supra closed. Then β-( ST, PT)*-cl(A)[β-bi-cl(A)] ( ST, PT)*-cl(A)[bi-cl(A)] U. Hence A is gβ-(τ,ατ)*[gβ-bi ]-supra closed Example 4.8: Let X={1,2,3,4} with τ={φ,{1,2,},{1,2,4},{1,2,3},x}.the set{1,4} is gβ-(τ,ατ)*-supra closed but not g- ( ST, PT)-supra closed. Example 4.9: Let X={1,2,3,4} with τ={φ,{1},{2},{1,2,},{1,2,4},{1,2,3},x}.the set {1,3}is gβ-bi-supra closed but not g- bi-supra closed Proposition 4.10: Every αg-( ST, PT)*[αg-bi ]-supra closed is gβ-( ST, PT)*[gβ-bi ]-supra closed but not conversely. Let A X be a αg-( ST, PT)*[αg-bi ]-supra closed and Let A U where U is (ST, PT)*[bi ] supra open set. Since A is a αg-( ST, PT)[αg-bi ]-supra closed. Then β(st, PT)*-cl(A)[β-bi-cl(A)] α-( ST, PT)*- cl(a)[α-bi-cl(a)] U. Hence A is gβ-( ST, PT)*[gβ-bi ]-supra closed. Example 4.11: Let X={1,2,3,4} with τ={φ,{1,2},{1,2,4},{1,2,3},x}.the set {1,3} is gβ-( ST, PT)*-supra closed set but not αg-( ST, PT)*-supra closed set. Example 4.12:

5 243 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4} with τ={φ,{3},{1,2},{1,2,3},x}. The set {1,3}is gβ-bi-supra closed but not αg-bi-supra closed. Proposition 4.13: Every gα-(st, PT)*[gα-bi]-supra closed is gs-(st, PT)*[gS-bi]-supra closed but not conversely. Let A X be a gα-(st, PT)*[gα-bi]-supra closed and Let A U where U is α-( ST, PT)*[α-bi]-supra open set. Since A is a gα-(st, PT)*[gα-bi]-supra closed. Then S-( ST, PT)*-cl(A)[S-bi-cl(A)] α- ( ST, PT)*-cl(A)[α-bi-cl(A)] U. Hence A is gs-( ST, PT)*[gS-bi]-supra closed. Example 4.14: Let X={1,2,3,4} with τ={φ, {1},{2},{1,2}.{1,3}.{1,2,3},{1,2,4}, X}. The set {1,3} is gs-( ST, PT)* supra closed but not gα-( ST, PT)*-supra closed. Example 4.15: Let X={1,2,3,4} with τ={φ, {1},{3},{1,3},{1,3,4},X}.The set {1,4} is gs-bi-supra closed but not gα-bisupra closed. Proposition 4.16: Every gs-( ST, PT)*[gS-bi]-supra closed is gβ-( ST, PT)*[ gβ-bi ]-supra closed but not conversely. Let A X be a gs-( ST, PT)*[gS-bi]-supra closed and Let A U where U is (ST, PT)*[bi ] supra open set. Since A is gs-( ST, PT)*[gS-bi]-supra closed.then β-( ST, PT)*-cl(A)[β-bi-cl(A)] S-( ST, PT)*- cl(a)[s-bi-cl(a)] U. Hence A is gβ-( ST, PT)*[ gβ-bi ]-supra closed. Example 4.17: Let X={1,2,3,4} with τ ={φ,{1,2},{1,2,3},{1,2,4},x}.the set {2,3} is gβ-( ST, PT)*-supra closed but not gs-( ST, PT)*-supra closed. Example 4.18: Let X={1,2,3,4} with τ ={ φ,{3},{1,2},{1,2,3},x }.The set {1,3} is gβ-bi-supra closed but not gs-bi-supra closed Proposition 4.19: Every αg-( ST, PT)*[αg-bi ]-supra closed is gs-( ST, PT)*[gS-bi]-supra closed but not conversely. Let A X be a αg-( ST, PT)*[αg-bi ]-supra closed and Let A U where U is( ST, PT)*[bi ]-supra open set. Since A is αg-( ST, PT)*[αg-bi ]-supra closed. Then S-( ST, PT)*-cl(A)[ S-bi- cl(a)] α-( ST, PT)*- cl(a)[α-bi-cl(a) ] U. Hence Ais is gs-( ST, PT)*[gS-bi]-supra closed. Example 4.20: Let X={1,2,3,4} with τ={φ,{1}, {2},{1,2},{1,3},{1,2,3},{1,2,4} X}.The set {1,3} is gs-( ST, PT)*-supra closed but not αg-( ST, PT)*-supra closed. Example 4.21: Let X={1,2,3,4} with τ={φ, {1},{3},{1,3},{1,3,4},X}.The set{1,4} is gs-bi-supra closed but not αg-bisupra closed. Proposition 4.22: Every αg-( ST, PT)*[ gα-bi]-supra closed is gp-( ST, PT)*[ gp-bi]-supra closed but not conversely. Let A X be an αg-( ST, PT)*[ gα-bi]-supra closed and Let A U where U is (ST, PT)*[bi]-supra open set. Since A is αg-( ST, PT)*[ gα-bi]-supra closed.then P(ST, PT)*-cl(A)[P-bi-cl(A)] α(st, PT)*-cl (A)[α-bi-cl(A)] U. Hence A is gp-( ST, PT)*[ gp-bi]-supra closed. Example 4.23:

6 244 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4} with τ={φ, {1 2}, X}.The set {2,3} is gp-( ST, PT)*[ gp-bi]-supra closed but not αg- ( ST, PT)*[ gα-bi]-supra closed. Proposition 4.24: Every αg-( ST, PT)*-supra closed is Pg-( ST, PT)*-supra closed but not conversely. Let A X be an αg-( ST, PT)*-supra closed and Let A U where U is ( ST, PT)*-supra open set. Since A is αg-( ST, PT)*-supra closed. Then P- (ST, PT)*-cl(A) α-( ST, PT)*-cl(A) U. Hence A is Pg- ( ST, PT)*-supra closed. Example 4.25: Let X={1,2,3,4} with τ={φ, {1 2}, X}.The set {3,4} is Pg-( ST, PT)*-supra closed but not αg- ( ST, PT)*-supra closed. Proposition 4.26: Every αg-( ST, PT)*-supra closed is P-( ST, PT)*-supra closed but not conversely. Let A X be an αg-( ST, PT)*-supra closed and Let A U where U is ( ST, PT)*-supra open set. Since A is αg-( ST, PT)*-supra closed. Then P- (ST, PT)*-cl(A) α-( ST, PT)*-cl(A) U. Hence A is P- ( ST, PT)*-supra closed. Example 4.27: Let X={1,2,3,} with τ={ φ, {1}, {3}, {1,3}, {1,3,4},X}.The set {1,3} is P-( ST, PT)*-supra closed but not αg-( ST, PT)*-supra closed. Proposition 4.28: Every gα-( ST, PT)*[gα-bi]-supra closed is P-( ST, PT)*[P-bi]-supra closed but not conversely.. Let A X be an gα-( ST, PT)*[gα-bi]-supra closed and Let A U where U is α( ST, PT)*[α-bi]-supra open set. Since A is gα-( ST, PT)*[gα-bi]-supra closed. Then P- (ST, PT)*-cl(A)[P-bi-cl A ] α-( ST, PT)*- cl(a)[α-bi-cl(a)] U. Hence A is P-( ST, PT)*[P-bi]-supra closed. Example 4.29: Let X={1,2,3,4} with τ={φ, {1},{3},{1,3},{1,3,4},X}. The set {2,4}is P-( ST, PT)*[P-bi]-supra closed but not gα-( ST, PT)*[gα-bi]-supra closed. Proposition 4.30: Every ( ST, PT)*-supra closed is W-( ST, PT)*-supra closed but not conversely. Let A X be a ( ST, PT)*-supra closed and Let A U where U is ( ST, PT)*-supra open set. Since A is ( ST, PT)*-supra closed. Then S- (ST, PT)*-cl(A)=A U. Hence A is W-( ST, PT)*-supra closed. Example 4.31: Let X={1,2,3,4}with τ={ φ,{3},{1,2},{3,4},{1,2,3},x}.the set {3} is W-( ST, PT)*-supra closed but not ( ST, PT)*-supra closed. Proposition 4.32: Every g-bi-supra closed is W-bi-supra closed but not conversely. Let A X be g-bi-supra closed and Let A U where U is bi-supra open set.since A is g-bi-supra closed. Then bi-cl(a) = A U.Since every supra open set is semi supra open set. Then U is semi supra open set. Hence A is W-bi-supra closed Example.4.33:

7 245 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4} with τ={φ,{1},{2},{1,2},{1,3}, {1,2,3}, {1,2,4},X}.The set {3,4} is W-bi-supra closed but not g-bi-supra closed. Proposition 4.34: Every W-( ST, PT)*[W-bi]-supra closed is Sg-( ST, PT)*[Sg-bi]-supra closed but not conversely. Let A X be a W-( ST, PT)*[W-bi]-supra closed and Let A U where U is( ST, PT)*[bi]-supra open set. Since A is W-( ST, PT)*[W-bi]-supra closed. Then S-( ST, PT)*-cl(A)[S-bi-cl(A)] ( ST, PT)*-cl(A)[bicl(A)] U. Hence A is Sg-( ST, PT)*[Sg-bi]-supra closed. Example 4.35: Let X={1,2,3,4} with τ={φ,{3},{1,2},{1,2,3},x}.the set {3} is Sg-( ST, PT)*-supra closed but not W- ( ST, PT)*-supra closed. Example 4.36: Let X={1,2,3,4}with τ={φ,{1},{2},{1,2},{1,3},{1,2,3},{1,2,4}, X}. The set {2,3} is Sg-bi-supra closed but not W-bi-supra closed. Proposition 4.37: Every Sg-(ST,PT)*[Sg-bi]-supra closed is β-(st,pt)*[β-bi]-supra closed but not conversely. Let A X be a Sg-(ST,PT)*[Sg-bi]-supra closed and Let A U where U is (ST,PT)*[bi]-supra open set. Since A is -(ST,PT)*[Sg-bi]-supra closed Then β-(st,pt)*[β-bi]-cl(a) S-(ST,PT)*[S-bi]-cl(A) U Hence A is β-(st,pt)*[β-bi]-supra closed. Example 4.38: Let X={1,2,3,4}with T={φ,{1,2},{1,2,4},{1,2,3},X}.The set {1,3} is β-(st,pt)*[β-bi]-supra closed but not Sg-(ST,PT)*[Sg-bi]-supra closed. Proposition 4.39: Every Sg-(ST,PT)*[Sg-bi]-supra closed is gβ-(st,pt)*[gβ-bi]-supra closed but not conversely. Let A X be a Sg-(ST,PT)*[Sg-bi]-supra closed and Let A U where U is (ST,PT)*[bi]-supra open set. Since A is -(ST,PT)*[Sg-bi]-supra closed Then β-(st,pt)*[β-bi]-cl(a) S-(ST,PT)*[S-bi]-cl(A) U Hence A is gβ-(st,pt)*[gβ-bi]-supra closed. Example 4.40: Let X={1,2,3,4}with T={φ,{1,2},{1,2,4},{1,2,3},X}.The set {2,3} is gβ-(st,pt)*[gβ-bi]-supra closed but not Sg-(ST,PT)*[Sg-bi]-supra. Proposition 4.41: Every (ST,PT)*[bi]-supra closed is α-(st,pt)*[α-bi]-supra closed but not conversely. Let A X be a (ST,PT)*[bi]-supra closed set, since A is (ST,PT)*[bi]-supra closed Then (ST,PT)*[bi]-cl(A)=A. We want to prove that (ST,PT)*[bi]-cl((ST,PT)*[bi]-int((ST,PT)*[bi]-cl(A))) A.since (ST,PT)*[bi]-int(A) A and A=(ST,PT)*[bi]-cl(A). Then (ST,PT)*[bi]-int((ST,PT)*[bi]- cl(a)) A (ST,PT)*[bi]-cl((ST,PT)*[bi]-int((ST,PT)*[bi]-cl(A))) (ST,PT)*-[bi]cl(A)=A (ST,PT)*[bi]-cl((ST,PT)*[bi]-int((ST,PT)*[bi]-cl(A))) A. Hence A is α-(st,pt)*[bi]-supra closed. Example 4.42: Let X={1,2,3,4}with T={φ,{1},{3},{1,3},{2,3},{1,2,3},X}.The set {3} is α-(st,pt)*-supra closed set but not (ST,PT)*-supra closed set. Example 4.43:

8 246 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4} with T={φ,{1},{3},{1,3},{1,3,4},X}. the set{4} is α-bi-supra closed set but not bi-supra closed set. Proposition 4.44: Every (ST,PT)*-supra closed is S-(ST,PT)*-supra closed but not conversely. Let A X be a (ST,PT)*-supra closed set, since A is (ST,PT)*-supra closed. Then (ST,PT)*-cl(A)=A. We want to prove that (ST,PT)*-int((ST,PT)*-cl(A))) A. since (ST,PT)*-int(A) A and A=(ST,PT)*- cl(a). Then (ST,PT)*-int((ST,PT)*-cl(A))) A.Hence A is S-(ST,PT)*-supra closed. Example 4.45: Let X={1,2,3,4}with T={φ,{3},{1,2},{3,4},{1,2,3},X}.The set{3} is S-(ST,PT)*-supra closed but not (ST,PT)*-supra closed. Proposition 4.46: Every S-(ST,PT)*[S-bi]-supra closed is β-(st,pt)*[β-bi]-supra closed but not conversely. Let A X be a S-(ST,PT)*[S-bi]-supra closed set such that (ST,PT)*[bi]-int((ST,PT)*[bi]-cl(A))) A.We want to prove that (ST,PT)*[bi]-int((ST,PT)*[bi]-cl((ST,PT)*[bi]-int(A))) A. Since (ST,)*[bi]- int(a) A and A=(ST,PT)*[bi]-cl(A). Then (ST,PT)*[bi]-int(A) A (ST,PT)*[bi]-cl((ST,PT)*[bi]-int(A)) (ST,PT)*[bi]-cl(A) (ST,PT)*[bi]-int((ST,PT)*[bi]-cl((ST,PT)*[bi]-int(A))) (ST,PT)*[bi]-int((ST,PT)*[bi]-cl(A)) A Therefore (ST,)*[bi]-int((ST,PT)*[bi]-cl((ST,PT)*[bi]-int(A))) A. Hence A is β-(st,)*[β-bi]-supra closed. Example 4.47 : Let X={1,2,3,4}with T={φ,{1,2},X}.The set {2,4} is β-(st,pt)*[β-bi]-supra closed but not S-(ST,PT)*[ S-bi]-supra closed. Proposition 4.48: Every gp-(st,pt)*[ɡp-bi]-supra closed is Pg-(ST,PT)*[ -bi]-supra closed but not conversely. Let A X be a gp-(st,pt)*[gp-bi]-supra closed and Let A U where U is (ST,PT)*[bi]-supra open set. Since A is -(ST,PT)*[ gp-bi]-supra closed and every (ST,PT)*[bi]-supra open set is P-(ST,PT)*[P-bi]-supra open set Then P-(ST,PT)*[P-bi]-cl(A) = A U Hence A is Pg-(ST, PT)*[Pg-bi]-supra closed set. Example 4.49 : Let X={1,2,3,4}with T={φ,{1,2},X}.The set {3,4} is Pg-(ST,PT)*[Pg-bi]-supra closed but not gp- (ST,PT)*[gP-bi]-supra closed. Proposition 4.50 : Every α-bi-supra closed is gα-bi-supra closed but not conversely. Let A X be a α-bi-supra closed and Let A U where U is α-bi-supra open set. Since A is α-bi-supra closed. Then α-bi-cl(a)=a U. Hence A is gα-bi-supra closed. Proposition 4.51: Every α-bi-supra closed is αg-bi-supra closed but not conversely Let A X be a α-bi-supra closed and Let A U where U is bi-supra open set. Since A is α-bi-supra closed. Then α-bi-cl(a)=a U. Hence A is αg-bi-supra closed Example 4.52:

9 247 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4}with T={φ,{3},{1,2},{3,4},{1,2,3},X}.The set{2} is gα-bi-supra closed[αg-bi-supra closed] but not α-bi-supra closed. Proposition 4.53: Every g-bi-supra closed is gs-bi-supra closed but not conversely Let A X be a g-bi-supra closed and Let A U where U is bi-supra open set. Since A is ɡ-bi-supra closed Then S-bi-cl(A) bi-cl(a) U. Hence A is gs-bi- supra closed. Example 4.54: Let X={1,2,3,4}with T={φ,{1},{2},{1,2},{1,3},{1,2,3},{1,2,4},X}.The set{1,3} is gs-bi-supra closed but not g-bi-supra closed. Proposition 4.55: Every Pg-bi-supra closed is gβ-bi-supra closed but not conversely. Let A X be a Pg-bi-supra closed and Let A U where U is P-bi-supra open set. Since A is -bi-supra closed Then β-bi-cl(a) P-bi-cl(A) U. Hence A is gβ-bi-supra closed. Proposition 4.56: Every gp-bi-supra closed is gβ-bi-supra closed but not conversely. Let A X be a gp-bi-supra closed and Let A U where U is bi-supra open set. Since A is gp-bi-supra closed.then β-bi-cl(a) P-bi-cl(A) U. Hence A is gβ-bi-supra closed. Example 4.57: Let X={1,2,3,4}with T={φ,{1},{3},{1,3},{2,3},{1,2,3},X}.The set {1,2} is gβ-bi-supra closed but not Pg-bi-supra closed[gp-bi-supra closed]. Proposition 4.58: Every S-bi-supra closed is Sg-bi-supra closed but not conversely Let A X be a S-bi-supra closed and Let A U where U is S-bi-supra open set Since A is S-bi-supra closed Then S-bi-cl(A)=A U. Hence A is Sg-bi-supra closed. Proposition 4.59: Every S-bi-supra closed is gs-bi-supra closed but not conversely Proof : Let A X be a S-bi-supra closed and Let A U where U is bi-supra open set. Since A is S-bi-supra closed Then S-bi-cl(A)=A U. Hence A is gs-bi-supra closed. Example 4.60: Let X={1,2,3,4}with T={φ,{1},{2},{1,2},{1,3},{1,2,3},{1,2,4},X}.The set {1} is Sg-bi-supra closed[gs-bi-supra closed] but not S-bi-supra closed. Proposition 4.61: Every β-bi-supra closed is gβ-bi-supra closed but not conversely Let A X be a β-bi-supra closed and Let A U where U is bi-supra open set. Since A is β-bi-supra closed Then β-bi-cl(a)=a U. Hence A is gβ-bi-supra closed. Example 4.62:

10 248 Tahh H. Jasim and Firas N. Jasim, 2014 Let X={1,2,3,4}with T={φ,{3},{1,2},{3,4},{1,2,3},X}.The set {3} is gβ-bi-supra closed but not β-biclosed. Proposition 4.63: Every P-bi-supra closed is β-bi-supra closed but not conversely : Let A X be P-bi-supra closed such that bi-cl(bi-int(a)) A.we want to prove that bi-int(bi-cl(biint(a)) A. since bi-int(a) A and bi-cl(a)=a. Then bi-int(a) A bi-cl(bi-int(a)) bi-cl(a) bi-cl(biint(a)) A bi-int(bi-cl(bi-int(a))) bi-int(a) A. Therefore bi-int(bi-cl(bi-int(a))) A. Hence A is β-bi-supra closed. Example 4.64:. Let X={1,2,3,4}with T={φ,{1},{3},{1,3},{1,3,4},X}.The set {3,4} is β-bi-supra closed but not P-bi-supra closed. Proposition 4.65: Every P-bi-supra closed is gβ-bi-supra closed but not conversely Let A X be a P-bi-supra closed and Let A U where U is bi-supra open set. Since A is P-bi-supra closed Then β-bi-cl(a) P-bi-cl(A) U. Hence A is gβ-bi- supra closed. Example 4.66: Let X={1,2,3,4}with T={φ,{1},{3},{1,3},{1,3,4},X}.The set {3,4} is gβ-bi-supra closed but not P-bisupra closed. Proposition 4.67: Every α-bi-supra closed is S-bi-supra closed but not conversely. Let A X be a α-bi- supra closed. Since A is α-bi-supra closed. Then bi-cl(bi-int(bi-cl(a)) A. we want to prove that bi-int(bi-cl(a)) A. Let G = bi-int(bi-cl(a)). Then G bi-cl(g) such that, bi-int(bi-cl(a)) bicl(bi-int(bi-cl(a))) A. Therefore bi-int(bi-cl(a)) A. Hence A is -bi-supra closed. Example 4.68: Let X={1,2,3,4}with T={φ,{3},{1,2},{1,2,3},X}.The set {3} is a S-bi-supra closed but not α-bi-supra closed.

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