IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 IJ----PPPeeerrrfffeeecccttt FFFuuunnnccctttiiiooo nnnsss BBBeeetttwwweeeeeennn BBBiiitttooo pppooo lllooo gggiiicccaaalll SSSpppaaaccceeesss Y. Y. Yousif and N. S. Jassim De partment of Mathe matics, College of Education- Ibn Al- Haitham, Unive rsity of Baghdad Abstract In this p aper we introduce a lot of concepts in bitop ological sp aces which are ij-ω- converges to a subset, ij-ω-directed toward a set, ij--closed functions, ij--rigid set, ij-- continuous functions and the main concept in this p aper is ij--p erfect functions between bitop ological sp aces. Several theorems and characterizations concerning these concepts are st udied. 1. Introduction and Preliminaries A set X with two top ologies 1 and 2 is called bitop ological sp ace [1] and is denoted by (X, 1, 2). The closure and interior of A in (X, i) is denoted by i-cl(A) and i-int(A), where i=1, 2. For ot her notions or not ations not defined here we follow closely R. Engelking [2]. De fini tion: 1.1. [3]. A filter  on a set X is a nonempty collection of nonemp ty subsets of X with the prop erties: (a) If F1, F2, then F1∩F2, (b) If F and FF*X, t hen F*. De fini tion: 1.2. [3]. A filter base  on a set X is a nonemp ty collection of nonemp ty subsets of X such that if F1, F2 then F3F1∩F2 for some F3. De fini tion: 1.3. [3]. If  and G are filter bases on X, we say that G is finer than  (written as  < G) if for each F, there is GG such that GF and that  meets G if F∩G for every F and GG. De fini tion: 1.4. [3]. A p oint x of a sp ace X is called a condensation p oint of the set AX if every nbd of the point x contains an uncountable subset of this set. Clearly the set of all condensation points of a set A is closed. De fini tion: 1.5. [4]. A subset of a sp ace X is called ω-closed if it contains all its condensation p oints. Also cl ω (A) will denote the intersection of all ω-closed sets which contains A. i.e., cl ω (A)=∩{F: F is ω-closed and AF}, then A is ω-closed iff A=cl ω (A). 2. IJ--Perfect Functions between Bitopological Spaces In this section a number of useful results about ij-ω-converges to a subset, ij-ω- directed toward a set, ij--closed functions, ij--rigid set, and ij--continuous functions are derived and used to obtain characterization theorem for an ij--p erfect functions between bitop ological sp aces. De fini tion: 2.1. A p oint x in bitop ological sp ace (X, 1, 2) is called an ij-ω-condensation p oint of a subset A of X iff for any i-op en nbd U of x, (j -cl ω (U))A. The set of all ij-- condensation p oints of A is called the ij-ω-closure of A and denoted by ij-ω-cl ω (A). A set AX is called ij-ω-closed if A=ij-ω-cl ω (A). De fini tion: 2.2. A p oint x in bitop ological sp ace (X, 1, 2) is called an ij-ω-condensation p oint of a filter base  on X if it is an ij-ω-condensation point of every number of . The set of all ij-ω-condensation p oints of  is called ij-ω-condensed of  and is denoted by ij-ω- cod. IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 De fini tion: 2.3. A filter base  on bitop ological sp ace (X, 1, 2) is said to be ij-ω-converges to a subset AX (written as   ij A) if for every i-op en cover A of A, there is a finite subfamily B A and F such that F{i-cl ω (B) : BB}. We say  ij-ω-converges to a p oint xX (written as   ij x) iff   ij {x} or equivalently , i-cl ω (U) of every i- op en nbd U of x contains some member of . The orem: 2.4. A p oint x in bitop ological sp ace (X, 1, 2) is an ij-ω-condensation of a filter base  on X if there exists a filter base * finer than  such that *  ij x. Proof: () Let x be an ij-ω-condensation p oint of a filter base  on X, then every i-op en nbd U of x, the j -ω-closure of U contains a member of  and thus contains a member of any filter base * finer than , so t hat *  ij x. () Sup p ose that x is not an ij-ω-condensation p oint of a filter base  on X, t hen there exists an i-op en nbd U of x such that j -ω-closure of U contains no member of . Denote by * the family of sets F*=F∩(Xj -cl ω (U)) for F, then the sets F* are nonemp ty . Also * is a filter base and indeed it is finer than , because given F1*=F1∩(Xj -cl ω (U)) and F2*=F2∩(Xj - cl ω (U)), there is an F3F1∩F2 and this gives F3*=F3∩(Xj -cl ω (U))F1∩F2∩(Xj - cl ω (U))=F1∩(Xj -cl ω (U))∩F2∩(Xj -cl ω (U)), by construction * not ij-ω-convergent to x. This is a contradiction, and thus x is an ij-ω-condensation p oint of a filter base  on X. De fini tion: 2.5. A filter base  on bitop ological sp ace (X, 1, 2) is said to be ij-ω-directed toward a set AX, written as     dij A, iff every filter base G finer than  has an ij-ω- condensation p oint in A. i.e., (ij-ω-codG)A. We write     dij x to mean     dij {x}, where xX. The orem: 2.6. Let  be a filter base on bitop ological sp ace (X, 1, 2) and a p oint xX, then   ij x iff     dij x. Proof: () If  does not ij--converge to x, then there exists a i-op en nbd U of x such that Fj -cl  (U), for all F. Then G={(X-i-cl  (U)F : F} is a filter base on X finer than , and clearly xij--codG. Thus  cannot be ij--directed towards x. () Clear. De fini tion: 2.7. A function f : (X, 1, 2)(Y, S1, S2) is said to be ij--p erfect if for each filter base  on f(X), ij--directed towards some subset B of f(X), t he filter base f -1 () is ij- -directed towards f -1 (B) in X. In the following theorem we show that only p oints of Y could be sufficient for the subset B in definition (2.7) and hence ij--direction can be replaced in view of theorem (2.4) by ij--convergence. The orem: 2.8. Let f : (X, 1, 2)(Y, S1, S2) be a function. Then the following are equivalent: (a) f is ij--p erfect. (b) For each filter base  on f(X), which is ij--convergent to a p oint y in Y, f - 1 ()    dij f -1(y ). (c) For any filter base  on X, ij--cod f ()f(ij--cod ). Proof: (a)(b) Follows from theorem (2.6). (b)(c) Let yij--cod f (). Then by theorem (2.4), there is a filter base G on f(X) finer than f () such that G  ij y . Let U ={f-1(G)F : G G and F}. T hen U is a filter base on X finer than f -1 (G). Since G    dij y , by theorem (2.6) and f is ij--p erfect, f - 1 (G)    dij f -1(y ). U being finer than f -1(G), we have f -1(y )(ij--cod U). It is then clear that f -1 (y )(ij--cod ). Thus yf(ij--cod ). (c)(a) Let  be a filter base on f(X) such that it is ij--directed towards some subset B of f(X). Let G be a filter base on X finer than f -1 (). Then f(G) is a filter base on f(X) finer than IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010  and hence B(ij--cod f(G)). Thus by (c) Bf(ij--codG) so that f -1 (B)(ij--cod G). This shows t hat f -1 () is ij--directed towards f -1 (B). Hence f is ij--p erfect. De fini tion: 2.9. A function f : (X, 1, 2)(Y, S1, S2) is called ij--closed if the image of each ij--closed set in X is ij--closed in Y. The orem: 2.10. A function f : (X, 1, 2)(Y, S1, S2) is ij--closed if ij--cl ω f(A)f(ij-- cl ω (A)), for each AX. Proof: Straightforward. The orem: 2.11. The ij--p erfect function f : (X, 1, 2)(Y, S1, S2) is ij--closed. Proof: Follows from theorem (2.10) and theorem (2.8 (a)(c)) by taking ={A}. De fini tion: 2.12. A subset A of bitop ological sp ace (X, 1, 2) is said to be ij--rigid in X if for each filter base  on X with (ij--cod)A=, there is Ui and F such that AU and j -cl  (U))F=, or equivalent, iff for each filter base  on X whenever A(ij--cod )=, then for some F, A(ij--cl  (F))=. The orem: 2.13. If a function f : (X, 1, 2)(Y, S1, S2) is ij--closed such that for each yY, f -1 (y ) is ij--rigid in X, t hen f is ij--p erfect. Proof: Let  be a filter base on f(X) such that   ij y in Y, for some yY. If G is a filter base on X finer than the filter base f -1 (), then f(G) is a filter base on Y, finer than . Since     dij y by theorem (2.4), yij--cod f(G), i.e., y{ij--clf(G) : GG}and hence y{f(ij--cl  (G) : GG} by theorem (2.10), since f is ij--closed. Then f -1 (y )ij-- cl  (G), for all GG. Hence for all Ui with f -1 (y )U, j -cl  (U)G, for all GG. Since f -1 (y ) is ij--rigid, it then follows that f -1 (y )(ij--codG). Thus f -1 ()    dij f -1(y ). Hence by theorem (2.8 (b)(a)), f is ij--p erfect. De fini tion: 2.14. A function f : (X, 1, 2)(Y, S1, S2) is called ij--continuous if for any Si- op en nbd V of f(x), there exists a i-op en nbd U of x such that f(j -cl  (U))Sj -cl  (V). The orem: 2.15. If an ij--continuous function f : (X, 1, 2)(Y, S1, S2) is ij--p erfect t hen f is ij--closed and for each yY, f -1 (y ) is ij--rigid in X. Proof: By theorem (2.11) f an ij--p erfect function is ij--closed. To p rove the other part, let yY, and sup p ose  is a filter base on X such that (ij--cod )f -1 (y )=. Then yf(ij--cod ). Since f is ij--p erfect, by theorem (2.8 (a)(c)) yij--cod f(). Thus there exists an F  such that yij--cl  f(F). There exists an Si-op en nbd V of y such that Sj -cl  (V)f(F)=. Since f is ij--continuous, for each xf -1 (y ) we shall get a i-op en nbd Ux of x such that f(j - cl  (Ux)Sj -cl  (V)Y-f(F). Then f(j -cl  (Ux)f(F)=, so that j -cl  (Ux)F=. Then xij-- cl  (F), for all x f -1 (y ), so t hat f -1 (y )(ij--cl  (F))=, Hence f -1 (y ) is ij--rigid in X. From theorems (2.13) and (2.15) we obtain. Corollary: 2.16. An ij--continuous function f : (X, 1, 2)(Y, S1, S2) is ij--p erfect if f is ij--closed and for each yY, f -1 (y ) is ij--rigid in X. We show that the above theorem remains valid if ij--closedness of f is replaced by a st rictly weaker condition which we shall call weak ij--closedness of f. Thus we define as follows. De fini tion: 2.17. A function f : (X, 1, 2)(Y, S1, S2) is said to be weakly ij--closed if for every yf(X) and every i-op en set U containing f -1 (y ) in X, there exists a Si-op en nbd V of y such that f -1 (Sj -cl  (V))j -cl  (U). The orem: 2.18. The ij--closed function f : (X, 1, 2)(Y, S1, S2) is weakly ij--closed. Proof: Let yf(X) and let U be a i-op en set containing f -1 (y ) in X. By theorem (2.10) and since f is ij--closed, we have ij--cl  f(X-j -cl  (U))f[i-cl  (X-j -cl  (U)]. Now since yf[i- cl  (X-j -cl  (U)], yij--cl  f(X-j -cl  (U)) and thus there exists an Si-op en nbd V of y in Y such that Sj -cl  (V)f(X-j -cl  (U))= which imp lies that f -1 (Sj -cl  (V))(X-j -cl  (U))=, i.e., f -1 (Sj -cl  (V))j -cl  (U), and thus f is weakly ij--closed. IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 The converse of the above theorem is not true, which is shown in the next examp le. Example: 2.19. Let 1, 2, S1 and S2 be any top ologies and f : (X, 1, 2)(Y, S1, S2) be a constant function, then f is weakly ij--closed for i, j=1 and 2 (ij). Now, let X=Y=IR. If S1 or S2 is the discrete top ology on Y, then f : (X, 1, 2)(Y, S1, S2) given by f(x)=0, for all xX, is neither 12--closed nor 21--closed, irresp ectively of the top ologies 1, 2 and S2 (or S1). The orem: 2.20. Let f : (X, 1, 2)(Y, S1, S2) be ij--continuous. Then f is ij--p erfect if (a) f is weakly ij--closed, and (b) f -1 (y ) is ij--rigid, for each yY. Proof: Sup p ose f : (X, 1, 2)(Y, S1, S2) is an ij--continuous function satisfy ing the conditions (a) and (b). To p rove that f is ij--p erfect we have to show in view of theorem (2.13) that f is ij--closed. Let yij--cl  f(A), for some non-null subset A of X, but yf(ij-- cl  (A). T hen Β ={A} is a filter base on X and (ij--codΒ)f -1 (y )=. By ij--rigidity of f - 1 (y ), there is a i-op en set U containing f -1 (y ) such that j -cl  (U)A=. By weak ij-- closedness of f, there exists an Si-op en nbd V of y such that f -1 (Sj -cl  (V))j -cl  (U), which imp lies that f -1 (Sj -cl  (V))A=, i.e., (Sj -cl  (V))f(A)=, which is imp ossible since yij-- cl  f(A). Hence yf(ij--cl  (A)). So f is ij--closed. From theorems (2.18) and (2.20) we get. Corollary: 2.21. Let f : (X, 1, 2)(Y, S1, S2) be an ij--continuous function. Then f is ij-- p erfect if (a) f is weakly ij--closed, and (b) f -1 (y ) is ij--rigid, for each yY. De fini tion: 2.22. A subset A in bitop ological sp ace (X, 1, 2) is called ij--set in X if for each i-op en cover Α of A, there is a finite subcollection Β of Α such that A{j -cl  (U) : BΒ}. The orem: 2.23. A subset A of a bitop ological sp ace (X, 1, 2) is an ij--set if for each filter base  on A, (ij--cod)A. Proof: () Clear. () Let Α be a i-op en cover of Α such that the j --closed of the union of any finite subcollection of Α is not cover A. Then ={A\j -cl  X(Β UΒ) : Β is finite sub collection of Α} is a filter base on A and (ij--cod)A=. This contradiction yields that A is ij--set. The orem: 2.24. If f : (X, 1, 2)(Y, S1, S2) is ij--p erfect and BY is an ij--set in Y, then f -1 (B) is an ij--set in X. Proof: Let  be a filter base on f -1 (B), then f() is a filter base on B. Since B is an ij--set in Y, Bij--codf() by theorem (2.23). By theorem (2.8 (a)(c)), Bf(ij--cod()), so that f -1 (B)ij--cod(). Hence by theorem (2.23), f -1 (B) is an ij--set in X. The converse of the above theorem is not true, is shown in the next examp le. Example: 2.25. Let X=Y=IR, 1 and 2 be the cofinite and discrete top ologies on X and S1, S2 resp ectively denote the indiscrete and usual top ologies on Y. Sup p ose f : (X, 1, 2)(Y, S1, S2) is the identity function. Each subset of either of (X, 1, 2) and (Y, S1, S2) is a 12--set. Now, any non-void finite set AX is 12--closed in X, but f(A) (i.e., A) is not 12--closed in Y (in fact, t he only 12--closed subsets of Y are Y and ). The theorem (2.24) and the above examp le suggest the definition of a st rictly weaker version of ij--p erfect functions as given below. De fini tion: 2.26. A function f : (X, 1, 2)(Y, S1, S2) is said to be almost ij--p erfect if for each ij--set K in Y, f -1 (K) is an ij--set in X. By analogy to theorem (2.13), a sufficient condition for a function to be almost ij-- p erfect, is p roved as follows. The orem: 2.27. Let f : (X, 1, 2)(Y, S1, S2) be any function such that IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 (a) f -1 (y ) is ij--rigid, for each yY, and (b) f is weakly ij--closed. Then f is almost ij--p erfect. Proof: Let B be an ij--set in Y and let  b e a filter base on f -1 (B). Now f() is a filter base on B and so by theorem (2.23), (ij--cod())B. Let y[ij--cod()]B. Sup p ose that  has no ij--condensation p oint in f -1 (B) so that (ij--cod())f -1 (y )=. Since f -1 (y ) is ij-- rigid, there exists an F and a i-op en set U containing f -1 (y ) such that Fi-cl  (U)=. By weak ij--closedness of f, there is a Si-op en nbd V of y such that f -1 (Sj -cl  (V))j -cl  (U) which implies that f -1 (Sj -cl  (V))F=, i.e., Sj -cl  (V)f(F)=, which is a contradiction. Thus by theorem (2.23), f -1 (B) is an ij--set in X and hence f is almost ij--p erfect. We now give some app lications of ij--p erfect functions. The following characterization theorem for an ij--continuous function is recalled to this end. The orem: 2.28. A function f : (X, 1, 2)(Y, S1, S2) is ij--continuous if f(ij--cl  (A))ij- -cl  f(A), for each AX. Proof: () Sup p ose that xij--cl  (A) and V is Si-op en nbd of f(x). Since f is ij-- continuous, there exists a i-op en nbd U of x such that f(j -cl  (U))Sj -cl  (V). Since j - cl  (U)A, then Sj -cl  (V)f(A). So, f(x)ij--cl  f(A). This shows that f(ij-- cl  (A))ij--cl  f(A). () Clear. The orem: 2.29. Let f : (X, 1, 2)(Y, S1, R2) be ij--continuous and ij--p erfect. Then f -1 p reserves ij--rigidity . Proof: Let B be an ij--rigid set in Y and let  be a filter base on X such that f -1 (B)(ij-- cod())=. Since f is ij--p erfect and Bf(ij--cod())= by theorem (2.8 (a)(c)) we get B(ij--cod f())=. Now B being an ij--rigid set in Y, there exists an F such that Bij- -cl  f(F)=. Since f is ij--continuous, by theorem (2.28) it follows that Bf(ij--cl  (F))=. Thus f -1 (B)(ij--cl  (F))=. This p roves that f -1 (B) is ij--rigid. Re ferences 1. Kelly J. C. (1963), Bitop ological Sp aces, Proc. London M ath. Soc., 13,71-89. 2. Englking R., (1989), Out line of General Top ology , Amst erdam. 3. Bourbaki N., 1975, General Top ology , Part I, Addison-Wesly, Reding, mass. 4. Hdeib H. Z., (1982), ω-closed map p ings, Revist a Colombian a de M athematics, XVI, 65- 78. IHJPAS 2010) 2( 32المجلد مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة ثنائیةالتبولوجیة الفضاءات البین IJ--الدوال التامة من النمط و نیران صباح جاسم یوسف یعكوب یوسف ابن الھیثم، جامعة بغداد -قسم الریاضیات، كلیة التربیة الصةالخ التـي هیـة االقتـراب لمجموعـة جزئیـة في هذا البحث نحن قدمنا العدید من المفاهیم في الفضاءات التبولوجیة الثنائیـة -ijصـالبة مجموعـة مـن الـنمط ،,ij-الـدوال المغلقـة مـن الـنمط ،-ij-االتجاه المباشر لمجموعة من الـنمط، -ij-من النمط -مــن الـنمط دوال المسـتمرة، الـij--، والمفهــوم الرئیسـي فــي هـذا البجـث هــو الـدوال التامــة مـن الــنمطij-- بـین الفضــاءات .هیم درستاكذلك العدید من المبرهنات و الممیزات المتعلقة بهذه المف. التبولوجیة الثنائیة IHJPAS