II. Pre- Operator Compact Space. Hanan. K. Moussa (Department of Mathimatics, College of Education, AL-Mustansirya University, Iraq)

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 11, Issue 6 Ver. II (Nov. - Dec. 2015), PP Haa. K. Moussa (Departmet of Mathimatics, College of Educatio, AL-Mustasirya Uiversity, Iraq) Abstract: The mai object of this paper to itroduce T-pre-compact Ad a good pre operator. Key words: T-pre compact, good pre-operator. I. Itroductio I 1979 Kasahara [1] itroduce the cocept of operator associated with a topology Γ of a space X as amap from P(X) to P(X) such that u α(u) for every u Γ. Ad itroduced the cocept of a operator compact space o a topological space(x,γ) as a subset A of X is α-compact if for every ope coverig of A tyere exists a fiite sub collectio {c 1,c 2,,c } of such that A i=1 α(c i ). I 1999 Rosas ad Vielma[3] modified the defiitio by allowig the operator α to be defied i P(X) as a map α from Γ to P(X).Ad properties of α- compact spaces has bee i vestigated i [ 1,3 ].Ad [ 4] gives some theorems about α-compact. I 2013 Masur ad Moussa [2] itroduce the cocept of a operator T o pre-ope set i topological space(x, Γ pre ) amely T- pre-operator ad studied some of their properties. I this paper we itroduce the cocept of T-pre-ope set with compact Asubset A of X is called T-pre-compact if for ay T-pre-ope cover {U : } of A, has a fiite collectio that covers A ad A T(U ).I 2 Usig the pre-operator T,we itroduce the cocept pre-operator compact space, good preoperator. Ad we study some of their properties ad optaied ew results. I 3 we itroduce some properties i about pre-operator compact space ad give some results i relatio to a pre-operator separatio axioms. II. 2.1 Defiitio: Let (X,,T) be a pre-operator topological A subset A of X is said to be pre-operator compact(t- Pre Compact) if for ay T-pre-ope cover {U : } of A, has a fiite collectio that covers A ad A T(U ) i 2.2 Defiitio: Let (X,,T) be a -operator topological A subset A of X is said to be -operator compact if for ay T--ope cover {U : } of A, has a fiite collectio that covers A ad A T(U ). i I the followig theorem, we preset the relatioship betwee T-pre compact ad T-compact spaces: 2.3 Theorem: Every T-pre compact space is T- compact 2.4 Propositio: Every T- pre compact space is T- compact 2.5 Theorem: The uio of two T-pre compact sets is T-pre- compact set. Suppose that (X,,T) be a pre-operator topological space ad A, B be two T-pre compact subsets of (X,,T) Let W {U : } be T-pre-ope cover of A B Sice A A B ad B A B DOI: / Page

2 Hece, W is T-pre-ope cover of A ad B Sice A is T-pre compact set Therefore, there exist a fiite subcover { U, U )} of W covers A i Hece A T(U ) i Also, sice B is T-pre compact set Therefore, there exist a fiite subcover { U, j1 T( U )} of W covers B jm Hece B m j1 T(U ) j m Therefore, A B k k1 Thus, A B is T-pre compact set. T(U ) i1 U i2 U j2,,,, U i U jm 2.6 Corollary: The fiite uio of pre-operator compact subsets of X is pre-operator compact. } such that {T( U ), T( U i1 ),, T( i2 } such that {T( U ), T( U j1 ),, j2 2.7 Defiitio: Let f : (X,,T) (Y,,L) be a fuctio. The two pre-operators T ad L are said to be good preoperators if f(t(f 1 (U))) L(U), for all U is L-pre-ope set i Y. 2.8 Propositio: If T ad L are good pre-operators, the the (T,L) pre-cotiuous image of T-pre compact space is T- compact. Suppose that f : (X,,T) (Y,,L) be (T,L) pre-cotiuous fuctio ad (X,,T) be T-pre compact Let W {A : } be L-ope cover of Y Hece, Y A Sice f is (T,L) pre-cotiuous fuctio Therefore, f 1 (W) {f 1 (A ) : } is T-pre-ope cover of X ad sice X is T-pre compact space Therefore, there exist a fiite subcover {f 1 (A 1 ), f 1 (A 2 ),, f 1 (A )}, such that {T(f 1 (A 1 )), T(f 1 (A 2 )),, T(f 1 (A ))} covers X 1 Thus, X T(f (A i)) 1 f(x) f T(f (A i)) 1 Y f T(f (A i)) Sice T ad L are good pre-operators, the: DOI: / Page

3 Y L(A i) Hece Y is T- compact 2.9 Theorem : If T ad L are good pre-operators, the the T-pre compact space is (T,L) pre-irresolute topological property. Suppose that (X,,T) be a pre-operator compact space ad (Y,,L) be a pre-operator topological space Let f : (X,,T) (Y,,L) be (T,L) pre-irresolute homeomorphism fuctio ad let W {A : } be L- pre-ope cover of Y Hece, Y A Sice f is (T,L) pre-irresolute cotiuous fuctio, therefore f 1 (W) {f 1 (A ) : } is T-pre-ope cover of X Thus, X 1 f (A ) Sice X is T-pre compact space Hece there exist a fiite subcover {f 1 (A 1 ), f 1 (A 2 ),, f 1 (A )}, such that {T(f 1 (A 1 )), T(f 1 (A 2 )),, T(f 1 (A ))} covers X 1 Thus, X T(f (A i)) 1 f(x) f T(f (A i)) Sice f is o to, hece f(x) Y, ad thus: 1 Y f T(f (A i)) Sice T ad L are good pre-operators Hece, by defiitio (3.4.9) f(t(f 1 (A i ))) L(A i ) Therefore, Y L(A i) Hece, (Y,,L) is L-pre compact III. T-pre-compact & T-pre-separatio axioms 3.1 Theorem : If T is a pre-regular pre-subadditive operator, the every T-pre-compact subset of T-pre-Hausdorff space is T-pre-closed. Suppose that (X,,T) be a T-pre-Hausdorff space Let F be T-pre-compact set i X ad let x F c Sice X is T-pre-Hausdorff space Hece, for each y F, here exist disjoit T-pre-ope sets U x, V y of the poits x ad y, respectively, such that T(U x ) T(V y ) The collectio {V y : y F} is T-pre-ope cover of F Sice F is T-pre-compact set Hece there exist a fiite subcover {V y1, V y2,, V y } such that {T(V y1 ), T(V y2 ),, T(V y )} covers F Thus, F T(V yi ) DOI: / Page

4 Let V T(V yi ) ad U T(U xi) Sice F V Therefore, we have U is T-pre-ope set, x U ad U F c Hece, F c is T-pre-ope set Thus, F is T-pre-closed set. 3.2 Theorem: If T is a regular subadditive operator, the every T-pre-compact subset of T-Hausdorff space is T- closed. Suppose that (X,,T) be a operator Hausdorff space Let F be T-pre-compact set i X ad let x F c Sice X is T-Hausdorff space Hece, for each y F, there exist disjoit T-ope sets U x, V y of the poits x ad y, respectively, such that T(U x ) T(V y ) Sice, every T-pre-compact space is T-compact space Hece, F is T-compact set ad the collectio {V y : y F} is T-ope cover of F Sice F is T-compact Hece there exist a fiite subcover {V y1, V y2,, V y } such that {T(V y1 ), T(V y2 ),, T(V y )} covers F Thus, F T(V yi ) Let V T(V yi ) ad U T(U xi) Sice F V, the we have U is T-ope set, x U ad U F c Hece, F c is T-ope set Thus, F is T-closed set. 3.3 Propositio : If T is a pre-regular pre-subadditive operator, the every T-pre-compact subset of T-Hausdorff space is T- pre-closed. 3.4 Theorem: If T is pre-subadditive operator, the every T-pre-closed subset of T-pre-compact space is T-compact. Suppose that (X,,T) be a T-pre-compact space ad let F be T-pre-closed subset of X Let the collectio {A : } be T-ope cover of F, that is, F A Sice F is T-pre-closed subset of X Hece F c is T-pre-ope subset of X Sice, every T-ope set is T-pre-ope set Hece, the collectio {A : } is T-pre-ope cover ad {A : } {F c } s T-pre-ope cover of X Sice X is T-pre-compact space Therefore, there exist a fiite subcover { A,,, } such that {T( A ), T( A )} {F c } covers X Hece, X T(A i ) F c Sice T is pre-subadditive operator 1 A 2 A 1 A ),, T( 2 DOI: / Page

5 Therefore, T(A i ) F c is T-pre-ope But F X Hece, F T(A i ) Thus, F is T-compact. 3.5 Corollary : If T is pre-subadditive operator, the a T-closed subset of T-pre-compact space is T-pre-compact. 3.6 Corollary : If T is subadditive operator, the a T-closed subset of T-pre-compact space is T-compact. 3.7 Corollary : If T is subadditive operator, the a T-pre-closed subset of T-pre-compact space is T-pre-compact. 3.8 Corollary : Let (X,,T) be a T-pre-Hausdorff space ad T is a regular subadditive operator. If Y X is T-pre- T(U) ad T(U) compact, x Y c, the there exist T-pre-ope sets U ad V with x U, Y V, x T(V), y T(V). Let y be ay poit i Y Sice (X,,T) is a T-pre-Hausdorff space, therefore there exist two T-pre-ope sets V y, U x, such that T(U x ) T(V y ) The collectio {V y : y Y} is T-pre-ope cover of Y Now, sice Y is T-pre-compact, therefore there exists a fiite subcollectio {V y1, V y2,, V y } such that {T(V y1 ), T(V y2 ),, T(V y )} covers Y Let U (U xi ), V (V yi ) Sice U T(U xi ), for every i {1, 2,, } Therefore, T(U) T(V yi ), for every i {1, 2,, } Hece, T(U) T(V). I the followig theorem, we preset the relatio betwee T-pre-compact ad strogly T-pre-regular 3.9 Theorem: If T is a regular subadditive operator, the every T-pre-compact ad T-pre-Hausdorff space is strogly T-pre-regular Suppose that (X,,T) be a T-pre-compact ad T-pre-Hausdorff space Let x X ad B be T-pre-closed subset of X, such that x B By corollary (3.4.20), B is T-pre-compact subset of T-pre-Hausdorff space ad by theorem (3.4.14) B is T-preclosed set Hece, (X,,T) is strogly T-pre-regular 3.10 Theorem: If T is a regular subadditive operator, the every T-pre-compact ad T-pre-Hausdorff space is T-preregular Suppose that (X,,T) be a T-pre-compact ad T-pre-Hausdorff space Let F be T-closed subset of X ad x X, such that x F Sice X is T-pre-Hausdorff space Hece, for each y F, there exist T-pre-ope sets U x, V y, such that x U x, y V y ad T(U x ) T(V y ) The collectio {V y : y Y} is T-ope cover of F By corollary (3.4.19) Thus F is T-compact set DOI: / Page

6 Therefore, there exists a fiite subcover {V y1, V y2,, V y } such that {T(V y1 ), T(V y2 ),, T(V y )} covers F ad F T(V yi ) Let V (V yi ) ad U (U xi ), The, x U ad U, V are disjoit T-pre-ope sets, such that x U, F V ad T(U) T(V) Hece, (X,,T) is T-pre-regular 3.11 Theorem: If T is a regular subadditive operator, the every T-pre-compact ad T-pre-Hausdorff space is strogly T-pre-T Theorem: If T is a regular subadditive operator, the every T-pre-compact ad T-pre-Hausdorff space is T-pre-T Theorem: If T is a regular subadditive operator, the every T-pre-compact ad T-pre-Hausdorff space is T-preormal Suppose that (X,,T) be a T-pre-compact ad T-pre-Hausdorff space Let E, F be a pair of disjoit T-closed subsets of X Let x F ad by theorem (3.4.21), there exist two T-pre-ope sets U x, V E, such that x U x, E V E ad T(U x ) T(V E ) The collectio {U x : x F} be T-pre-ope cover of F Sice F is T-closed subset of T-pre-compact space ad by corollary (3.4.19), F is T-compact Hece, there exist a fiite subcollectio {U x1, U x2,, U x } such that {T(U x1 ), T(U x2 ),, T(U x )} covers F ad F T(U xi) Let U (U xi ) ad V (V Ei ) The, U ad V are disjoit T-pre-ope sets, such that F U, E V ad T(U) T(V) Hece, X is T-pre-ormal Refereces [1]. S. Kasahara "operatio-compact spaces ", Mathematic Japoica, 24(1979), [2]. N. G. Masour,H. K. Moussa "T-pre-operators",IOSR-JM,5(2013), [3]. E.Rosas,J.Vielma "operator compact ad operator coected spaces",scietiae Mathematicas (2) (1999), [4]. E.Rosas,J.Vielma "operator compactificatio of Topological spaces", Divulgacioes Mathematicas, vol.8,no.2 (2000), DOI: / Page