Journal of the Egyptian Mathematical Society (2013) 21, 318-323

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

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ORIGINAL ARTICLE

New types of generalized closed sets in bitopological spaces

H.M. Abu-Donia

Department of Mathematics, Faculty of Science, Zagazig University, Egypt

Received 7 March 2012; revised 1 October 2012; accepted 14 March 2013 Available online 4 May 2013

KEYWORDS

j-Near open sets;

j-Near continuous functions;

Binormal;

Almost binormal and mild binormal

Abstract In this paper, we introduce a new type of closed sets in bitopological space (X, t1; t2), used it to construct new types of normality, and introduce new forms of continuous function between bitopological spaces. Finally, we proved that the our new normality properties are preserved under some types of continuous functions between bitopological spaces.

MSC: 54C08 54C10 54C20

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1. Introduction and preliminaries

The concepts of regular closed, generalized closed (briefly, g-closed), preopen, regular generalized closed (briefly, rg-closed), and generalized preclosed (briefly, gp-closed) sets have been introduced and investigated in [1-5]. The concepts of preopen sets and regular open sets have been extended to bitopological spaces [6] called i/-preopen and «/-regular open respectively. The mild normality and almost normality have been introduced in [7]. A weak form of normal spaces has been introduced in [8] called mildly normal spaces. In [9], the author used the preopen sets to define prenormal spaces, recently, in [10] the author have continued the study of further properties of prenormal spaces and also defined and investigated mildly p-normal (resp. almost p-normal) spaces which are generaliza-

E-mail address: donia_1000@yahoo.com

Peer review under responsibility of Egyptian Mathematical Society.

tion of both mildly normal (resp. almost normal) spaces and p-normal spaces. The concept of generalized preregular closed (briefly, gpr-closed) sets has been introduced in [11]. The concept of binormal spaces has been introduced in [12]. In [13,14] extended the concepts of g-closed, gp-closed and rg-closed sets, mildly normal and almost normal spaces to bitopological spaces.

In this paper, we extend the concept of gpr-closed sets to bitopological spaces (X, t1, t2) called i/-gpr-closed sets. Also, we construct a new types of normality in bitopological spaces based on «/-preopen sets called prebinormal, almost prebinor-mal and mildly prebinormal. We use the class of i/-gpr-closed sets to characterization these types of normality and construct new types of continuous functions. We prove that the introduced binormality properties are preserved under some types of continuous functions.

Throughout this paper, the following abbreviations will be adopted: Let A be a subset of a bitopological space (X, t1, t2), the interior (resp. closure) of A with respect to topology t (i = 1, 2) will be denoted by int(A) (resp. cl(A)). We denote the set of all closed sets with respect to the topology t by i-C(X).

In what follows, let i, / 2 {1,2} and i „/

1110-256X © 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.Org/10.1016/j.joems.2013.03.005

Definition 1.1 [6]. A subset A of a bitopological space (X, t1; s2) is said to be

(1) ¿¿/-preopen if A C int,(clj(A)).

(2) ¿¿/-regular open if A = inti(clj(A)).

The complement of ¿¿/-preopen (resp. ¿/-regular open) set is called ¿¿/-preclosed (resp. ¿¿/-regular closed) set. We denote the set of all ¿j-preopen (resp. ¿¿/-preclosed, ¿¿/-regular open and ij-regular closed) sets by ij-PO(X) (resp. ij-PC(X), ij-RO(X) and ij-RC(X)).

Definition 1.2 [6]. For any bitopological space (X, s1,s2) and A c X, ¿¿/-preinterior (resp. ¿¿/-preclosure) of A is denoted by pmtj(A) (resp. pdy(A)) and defined as

pin/A) = UfF C X: F 2 ij - PO(X), F C A} (resp. pclj(A) = TfFC X: F 2 ij - PC(X), F D A}).

Definition 1.3. A subset A of a bitopological space (X, s1, s2) is said to be

(1) ¿¿/-generalized closed [14] (briefly, ¿¿/-g-closed) if A c U,U 2 st ) clj(A) c U.

(2) ¿¿/-regular generalized closed [13] (briefly, ¿¿/-rg-closed) if A c U, U 2 ij-RO(X) ) clj(A) c U.

(3) ¿¿/-generalized preclosed [14] (briefly, ¿¿/-gp-closed) if A C U, U 2 Si ) pclji(A) C U.

Definition 1.4. A subset A of a bitopological space (X, t1; x2) is said to be //-generalized preregular closed (briefly, ij-gpr-closed) if A ç U,U 2 ij-RO(X) ) pclji(A) ç U.

The complement of /j-g-closed (resp. /j-rg-closed, ij-gp-closed and /j-gpr-closed) set is called /j-g-open (resp. /j-rg-open, /j-gp-open and /j-gpr-open) set and defined in the following lemma.

Definition 1.4 is a particular case of Definition 8 from Noiri [15].

From Proposition 4 in [15], we obtain the following lemma.

need not be an /j-gpr-closed set as shown by the following example.

Example 1.1. Let X = {a, b, c,d}, = {X,/,{«}, {d}, {a, d}} and s2 = {X, /,{b},{a,b}}. We have {b, c}, {b,d} 2 21-GPRC (X) but {b, c} n {b, d} = {b} R 21-GPRC(X).

Proposition 1.1. The follow/rig d/agram shows the relatwnsMp between the above dfferent types of closed sets.

j-C(X) _ 5

ji-PC(X)

_ ij-GC(X) 6

_► ij-GPC(X)

* ij-RGC(X) 7

ij-GPRC(X)

where none of these implications ¿s revers^le as shown by the following example.

Example 1.2. Let X = {a,b, c,d}, s1 = {X, /,{a},{a,d}} and s2 = {X, /,{a, b},{c, d}}.

(Arrows 1, 5) {b} 2 12-GC(X) n 21-PC(X) but {b} R 2-C(X).

(Arrows 2, 6) {d} 2 12-RGC(X) n 12-GPC(X) but {d} R 12-GC(X) since there exist {a, d} 2 s1 containing {d} such that 2-cl({d}) = {c, d} /C {a, d}.

(Arrow 3) {a, b} 2 12-GPC(X) but {a, b} R 12-PC(X).

(Arrow 4) {a, b} 2 21-GPRC(X)but {a, b} R 21-GPC(X), since there exist {a, b} 2 s2 containing {a, b} such that 12-pcl({a, b}) = X CC {a, b}.

(Arrow 7) {c} 2 21-GPRC(X)but {c} R 21-RGC(X), since there exist {c, d} 2 21-RO(X) containing {c} such that 1-

cl({c, d}) = {c, d} /C {c}.

Remark 1.1. For any bitopological space (X,ti,s2) we note that:

(1) The classes ij-GC(X) and ji-PC(X) are independent

(2) The classes ij-RGC(X) and ij-GPC(X) are independent

Lemma 1.1. A subset A of a bUopologkal space (X, Xj, s2) /s:

(1) /j-g-open iff A □ F, F 2 /-C(X) ) mtj(A) □ F

(2) ij-rg-open iff A □ F, F 2 ij-RC(X) ) intj(A) □ F

(3) ij-gp-open iff A □ F, F 2 i-C(X) ) pintjfA) □ F

(4) ij-gpr-open iff A □ F, F 2 ij-RC(X) ) pintji(A) □ F

We denote the set of all ij-g-closed (resp. ij-g-open, ij-rg-closed, /j-rg-open, /j-gp-closed, /j-gp-open /j-gpr-closed and /j-gpr-open) sets by ij-GC(X) (resp.ij-GO(X), ij-RGC(X), ij-RGO(X), ij-GPC(X), ij-GPO(X), ij-GPRC(X) and ij-GPRO (X)).

The arbitrary union of ij-gpr-closed sets is an ij-gpr-closed set. But the intersection of two of /j-gpr-closed sets

The following example investigate the previous remark.

Example 1.3. Let (X, ti, t2) as in Example 1.2:

(1) {c, d} 2 21-RGC(X) but {c, d} R 21-GPC(X), Also, {c} 2 21-GPC(X)but{c} R 21-RGC(X).

(2) {a, c} 2 12-GC(X) but {a, c} R 21-PC(X), Also, {d} 2 21-PC(X)but{d} R 12-GC(X).

Theorem 1.1. For any bitopological space (X, tj, t2), A c X, the following are holds:

(1) If A 2 ij-GC(X) n ti then A 2 j-C(X).

(2) If A 2 ij-GPC(X) n St then A 2 ji-PC(X).

(3) If A 2 iJ-RGC(X) and st = /-RO(X) then A 2 i/-GC(X).

(4) If A 2 /-GPRC(X) and t = /-RO(X) then A 2 /-GPC(X).

(5) If A 2 /-GPRC(X) and/-C(X) = /i-PC(X) then A 2 /-RGC(X).

(6) If A 2 i/-GPC(X) and /-C(X) = /i-PC(X) then A 2 i/-GC(X).

Proof. Obvious. □

Theorem 1.2. For any bitopological space (X, sj, s2). If A 2 ij-GPRC(X) and A ç B ç ji-pcl(A), then B 2 ij-GPRC(X).

Proof. Let B ç U, U 2 ij-RO(X). Since A ç B and ij-GPRC(X), then ji-pcl(A) ç U. Since B ç ji-pcl(A), then we have ji-pcl(B) ç ji-pcl(A) ç U. Consequently B 2 ij-GPRC(X). □

Theorem 1.3. Let (Xj,xj,x2) and (X2, x\, s2) be two bitopological spaces. If A 2 ij-GPRO(XJ) and B 2 i*j*-GPRO(X2), then A x B 2 i • i*,j x j*-GPRO(XJ x X2).

Proof. Let A 2 ij-GPRO(Xi), B 2 i*j*-GPRO(X2) and W = A X B ç X1 X X2. Let F = F1 X F2 ç W,F 2 i X i*j X j*-RC(X1 x X2). Then, there are F1 2 ij-RC(X1) and F2 2 i*j*-RC(X2), Fi ç A, F2 ç B and so F1 çpintji(A) and F2 #pintj,i,(B). Hence Fi x F2 #pint j(A)x pintj,i,(B) = pintjXf ix? (A x B). Therefore A x B 2 i x i*, j x j*-GPRO (Xi x X2). ' □

2. Some types of ¿/-near continuous functions

In this section we introduce two types of continuous functions between bitopological spaces and study their properties.

Definition 2.1 [14]. A function f:(X,s1,s2) fi (Y, r1, r2) is called:

Proof. Straightforward. □

In Diagram 2.1, the arrows are not reversible as one may see the following examples:

Example 2.1. Let X = {a,b,c,d}, Y = {u, v,w}, ^ = {X, / ,{a},{a,d}} and t2 = {X,/,{a,b},{c,d}}. a1 = {Y,/,{u},{v, w}} and r2 = { Y,/,{v},{u, v}}. Letf:(X,t1,t2) fi (Y, r1,r2).

(arrows 1, 2) If f is defined by f(a) = u,f(b) = v and f(c) = f(d) = w. We have f is 12-precontinuous, but it is not 1-continuous. Since there exist {u} 2 1-C(Y) but f~1({u}) = {a} R 1-C(X). Also, f is 12-g-continuous, but it is not 2-continuous. Since there exist {u, w} 2 2-C(Y) such that r1({u, w}) = {a, c, d} R 2-C(X).

(arrows 3, 4) Iff is defined by f(a) = f(b) = u f(c) = v and f(d) = w. We have f is 12-gp-continuous, but it is not 12-g-continuous. Since there exist {w} 2 2-C(Y) but fT1({w}') = { d} R 12-GC(X). Also, f is not 21-precontinuous. Since there exist {u, w} 2 2-C(Y) such that r\{u,w}) = {a,b,d} R 12-PC(X).

(arrows 5, 6) Iff is defined by f(a) = f(d) = v f(b) = u and f(c) = w. We have f is 12-gp-continuous and 12-pre-gpr-continuous but it is not 12-pre-gp-continuous. Since there exist {v} 2 21-PC(Y) such that/L1({v}) = {a,d} R 12-GPC(X).

Remark 2.1. For any function f:(X,t1,t2) fi (Y, r1, r2), we note that:

(1) ij-g-continuous and ij-pre-gp-continuous are independent.

(2) /i- precontinuous and i/-g-continuous are independent.

(3) ij-gp-continuous and ij-pre-gpr-continuous are independent.

The following example justifies the previous remark.

Example 2.2 (i, ii) [resp. (iii, iv) and 2.4 (v, vi)] investigate Remark 2.1 (1) [resp. (2) and (3)].

(1) i/-precontinuous if "V 2 i-C(Y)iT1(V) 2 ij-PC(X)

(2) ij-g-continuous if "V2 j-C(Y),r\V) 2 ij-GC(X).

(3) j-gp-continuous if "V 2 j-C(Y), f\V) 2 ij-GPC(X).

(4) i-continuous if "V 2 i-C(Y), Г\У) 2 i-C(X).

Definition 2.2. A functionf:(X,s1,s2) fi (Y, r1, r2) is called:

(1) i/'-pre-gpr-continuous if "V2 ji-PC(Y), f^1(V) 2 ij-GPRC(X).

(2) j-pre-gp-continuous if "V 2 ji-PC(Y), f^1(V) 2 ij-GPC(X).

Theorem 2.1. The relationship between the previous concepts of continuity of functions between bitopological spaces are stated in the following diagram:

ji-pre continuous i

^-continuous _

„ у-до-continuous 3

^ îj-g-œntinuous

¿j-pre-^pr-continuouj

¿j-pre-pp-continuous

Example 2.2. Letf:(X,s1,s2) fi (Y,r1,r2) as in Example 2.1.

(i) Iff is defined by f(a) = v,f(b) = u and f(c) = f(d) = w. We have f is 12-g-continuous, but it is not 12-pre-gp-continuous. Since there exist {v} 2 21-PC(Y) such that r1((v}) = {a} R 12-GPC(X).

(ii) Iff is defined by f(a) = f(c) = v, f(b) = u and f(d) = w. We have f is 12-pre-gp-continuous, but it is not 12-g-continuous. Since there exist {w} 2 2-C(Y) such that ,T1((w}) = {d} R 12-GC(X).

(iii) Iff is defined by f(a) = f(b) = v f(c) = u and f(d) = w. We have f is 21-precontinuous, but it is not 12-g-contin-uous. Since there exist {w} 2 2-C(Y) such that ,T1({w}) = {d} R 12-GC(X).

(iv) If f is defined by f(a) = f(c) = w, f(b) = v and f(d) = u. We have f is 12-g-continuous, but it is not 21-precontin-uous. Since there exist {w} 2 2-C(Y) such that ,T1({w}) = {a,c} R 21-PC(X).

(v) If f is defined by f(a) = v, f(b) = f(d) = w and f(c) = u. We have f is 12-gp-continuous, but it is not 12-pre-gpr-

continuous. Since there exist {v} 2 21-PC(Y) such that f-1({v}) = {a} R 12-GPRC(X). (vi) If f is defined by f(a) = f(d) = w, f(b) = u and f(c) = v. We have f is 12-pre-gpr-continuous, but it is not 12-gp-continuous. Since there exist {w} 2 2-C(Y) such that f-i({w}) = {a,d} R 12-GPC(X).

Definition 2.3. A functionf:(X,si,s2) fi (Y, ri, r2) is called:

(1) ¿j-R-map if "V 2 ij-RO(Y)f-i(V) 2 ij-RO(X);

(2) ¿j-preirresolute if "V 2 ij-PC(Y), f-i(V) 2 ij-PC(X);

(3) ¿j-r-closed if "G 2 ij-RC(X), f(G) 2 ij-RC(Y);

(4) ¿j-pre-gp-closed if "G 2 ij-PC(X), f(G) 2 ji-GPC(Y);

(5) ¿j-pre-rgp-closed if "G 2 ij-PC(X),f(G) 2 ji-GPRC(Y);

Lemma 2.1. For any surjection function f:(X, sj, s2) fi (Y, <rj, a2) the following are equivalent.

(a) f is ij-pre-gp-closed function.

(b) For any B c Y, U 2 ij-PO(X) such that f-J(B) c U, there exist V 2 ji-GPO(Y) such that B c V and f-J(V) c U.

Proof. Necessity, Let B c Y, U 2 ij-PO(X) such that f-i(B) c U. Since f is ¿j-pre-gp-closed function, thenf(U) 2 ji-GPO(Y). Put V = f(U). Since f-i(B) c U, then B = f(T\B)) c f(U) = V and fW) = f-i(f(U)) c U. □

Sufficiency, Let G 2 ij-PO(X), f(G) □ F such that F 2 i-C(Y), then G □ f-i(F), Fc Y. This implies that there exist V 2 ji-GPO(Y) such that F c V and f-i(V) c G. Since V 2 ji-GPO(Y), F2 j-C(Y) and Fc V. Consequently, pint^V) D F. Since V c f(G), then F C pin/V) C pintfG)). This implies thatf(G) 2 ji-GPO(Y). Therefore, f is ¿j-pre-gp-closed function.

Lemma 2.2. For any surjection function f:(X, sj, s2) fi (-sj, s2) fi (Y, <rj, a2) the following are equivalent.

(a) f is ij-pre-rgp-closed function.

(b) For any B c Y, U 2 ij-PO(X) such that f-J(B) c U, there exist V 2 ji-GPRO(Y) such that B c V and f-J(V) c U.

Proof. Similar to Lemma 2.1. □

Theorem 2.2. Let f:(X,sj,s2) fi (Y, aj, a2) is ij-pre-gp-continuous

(resp. ij-pre-gpr-continuous) function and g: (Y, aj, a2) fi

(Z,gj,g2) is ij-preirresolute function, then gof:(X, sj,s2) fi (Z, gj, g2) is ij-pre-gp-continuous (resp. ij-pre-gpr-continuous)

Proof. Let V 2 ij-PC(Z), since g is ij-preirresolute, then

g-i(V) 2 ij-PC(Y). Since f is ¿j-pre-gp-continuous, then f-1(g~1(V)) = (g of)-i (V) 2 ji-GPC(X). Consequently, g o f ij-pre-gp-continuous. □

and almost prebinormal. We give a new characterization of these types of binormality by ij-gpr-open sets.

Definition 3.1 [12]. A bitopological space (X,xbx2) is said to be binormal if given disjoint subsets A, B, A 2 i-C(X) and B 2 j-C(X), there are disjoint subsets U, V such that U 2 j V 2 xi, A с U and B с V.

Definition 3.2. A bitopological space (X, s1, x2) is said to be pre binormal if given disjoint subsets A, B, A 2 i-C(X) and B 2 j-C(X), there are disjoint subsets U, V such that U 2 ji-PO(X),

V 2 ij-PO(X), A с U and B с V.

Theorem 3.1. For any bitopological space (X, Xj, x2), the following statements are equivalent:

(a) X is prebinormal;

(b) for any disjoint sets A 2 i-C(X) and B 2 j-C(X), there exist U 2 ij-GPRO(X), V 2 ji-GPRO(X) and U П V = / such that A с U and B с V

(c) for any A 2 i-C(X),G 2 Xj and G □ A, there exists U 2 ij-GPRO(X) such that A с Uсpcltj(U) с G.

Proof. (a) ) (b). Let A 2 i-C(X), B 2 j-C(X) and A П B = /. Since X is prebinormal, then there exist U 2 ji-PO(X), V 2 ij-PO(X) and U П V = / such that A с U and B с V, this follows that, there exist U 2 ij-GPRO(X), V 2 ji-GPRO(X) and U П V = / such that A с U and B с V.

(b) ) (c). Let A 2 i-C(X), G 2 xj and G □ A. Then, A 2 i-C(X), X\G 2 j-C(X), (X\G) П A = /. Then, there exist U 2 ij-GPRO(X), V 2 ji-GPRO(X)and U П V = / such that A с U and X\G с V. Since V 2 ji-GPRO(X), X\G 2 ji-RC(X) and X\G с V, then by using Lemma 1.1 (4) we have pt'nttj(V) □ X\G. UП V = / implies UПpintij(V) = /. Consequently, A с Uс X\pintij(V) с G, this follows that A с Uсpcl ij(U) с X\pintij(V) с G. Consequently, A с U с pcjU) с G.

(c) ) (a). Let A 2 i-C(X), B 2 j-C(X) and A П B = /. Then, A 2 i-C(X), X\B 2 Xj and A с X\B. Consequently, there exist G 2 ij-GPRO(X) such that A с G с pcjG) с X\B. Since A с G, A 2 ij-RC(X) and G 2 ij-GPRO(X) then, by using Lemma 1.1 (4) we have A сpinj(G). This follows that B с X\pclj(G) = pinty(Gc), pinj(G) 2 ji-PO(X), pintj(Gc) 2 ij-PO(X) and pintji(G) П pintij(Gc) = /. Put U = intj(cli(pint-ji(G))) and V = inti(clj(pintij(Gc))). Then U, V are disjoint, U 2 ji-PO(X) and V 2 ij-PO(X) Such that U □ A and

V □ B. □

Definition 3.3. A space (X, xb x2) is said to be almost prebinormal if given disjoint subsets A and B, A 2 i-C(X), B 2 ji-RC(X), there are disjoint subsets U and V such that U 2 ji-PO(X), V 2 ij-PO(X), A с U and B с V.

Theorem 3.2. For any bitopological space (X, xj, x2), the following statements are equivalent:

3. Some types of normality in bitopological spaces

In this section, we introduced three concepts of normality in bitopological spaces namely prebinormal, mild prebinormal,

(a) X is almost prebinormal;

(b) for each disjoint sets A 2 i-C(X) and B 2 ji-RC(X) there are disjoint subsets U 2 ij-GPO(X) and V 2 ji-GPO(X) such that A c U and B c V;

(c) for each disjoint sets A 2 i-C(X) and B 2 ji-RC(X) there are disjoint subsets U 2 ij-GPRO(X) and V 2 ji-GPRO(X) such that A ç Uand B ç V;

(d) for each A 2 i-C(X) and K 2 ji-RO(X), and K□ A there exists U 2 ij-GPRO(X) such that A ç U ç ij-pcl(U) ç к.

Proof. It is obvious that (a) ) (b) ) (c).

(c) ) (d). Let A 2 xc and K 2 ji-RO(X) and K□ A. This implies that A 2 xc and X\K 2 ji-RC(X) and (X\A) n A = / Then, there exists U 2 ij-GPRO(X) and V 2 ji-GPRO(X) such that A ç U, X\K ç V and U n V = /. Since V 2 ji-GPRO(X), X\K 2 ji-RC(X) and X\K ç V, then by Lemma 1.1 (4), we have pinjV) □ X\K. Since U n V = / implies U n pintij(V) = /. Consequently, A ç U ç X\pintj(V) ç K, this follows that A ç U ç pcly(U) ç X\pinty(V) ç K. Therefore, A ç U ç pclj(U) ç K

(d) ) (a). Let A 2 i-C(X), B 2 ji-RC(X) and A n B = /. This implies that A 2 i-C(X), X\B 2 ji-RO(X) and A ç X\B. Consequently, there exists U 2 ij-GPRO(X) such that A ç U ç pcliJ(U) ç X\B. Since A ç U, A 2 ij-RC(X)and U 2 ij-GPRO(X) then, by using Lemma 1.1 (4) we have A ç pintj(U). This follows that B ç X\pclj(U = pintij(Uc), pinti-j(U) 2 ji-PO(X), pintij(Uc) 2 ij-PO(X) and pinj(U) n pinti-/U) = /. Put G = intj(cli(pintji(U))) and H = inti(clj(pintij(Uc))). Then G, H are disjoint, G 2 ji-PO(X) and H 2 ij-PO(X) Such that G □ A and H □ B. □

Definition 3.4. A bitopological space (X, s1, s2) is said to be mildly prebinormal if given disjoint subsets A 2 ij-RC(X) and B 2 ji-RC(X), there are disjoint subsets U 2 ji-PO(X) and V 2 ij-PO(X) Such that A ç U and B ç V.

Theorem 3.3. For any bitopological space (X, xj, x2), the following statements are equivalent:

(a) X is mildly prebinormal;

(b) for any A 2 ij-RC(X), B 2 ji-RC(X) and A n B = / there are U 2 ij-GPO(X), V 2 ji-GPO(X) and U n V = / such that A ç U and B ç V;

(c) for any A 2 ij-RC(X), B 2 ji-RC(X) and A n B = / there are U 2 ij-GPRO(X), V 2 ji-GPRO(X) and U n V = / such that A ç U and B ç V;

(d) for any A 2 ij-RC(X), K 2 ji-RO(X) and A ç K there exists U 2 ij-GPO(X), such that A ç U ç pcly(U) ç K;

(e) for any A 2 ij-RC(X), K 2 ji-RO(X) and A ç K there exists U 2 ij-GPRO(X), such that A ç U çpclij(U) ç K;

Proof. Similar to that of Theorem 3.2. □ 4. Preservation theorems

In this section, we prove that the three types of binormality properties are preserved under some types of function between bitopological spaces.

Theorem 4.1. Iff:(X, xj, x2) fi (Y, aj, a2) is ij-pre-gp-closed, i-continuous, surjection and X is prebinormal then Y is also prebinormal.

Proof. Let A 2 i-C(Y), B 2 j-C(Y) and A n B = /, Since f is surjection i-continuous, then f^1(A) 2 i-C(X), f^1(B) 2 j-C(X) and fT1(Â) n r\B)= f\A n B) = /. Since X is prebinormal, there exist U 2 ji-PO(X), V 2 ij-PO(X) and U n V = / such thatf^1(A) ç U andf^1(B) ç V. Since f is ij-pre-gp-closed, by Lemma 2.1, there exist G 2 ij-GPO(Y) and H 2 ji-GPO(Y) such that A ç G, B ç H, f\G) ç U and f\H) ç V. Since U and V are disjoint, G and H are disjoint. Since G 2 ij-GPO(Y) and H 2 ji-GPO(Y), by Lemma 1.1 (3), then we have A çpint-ji(G), B ç pintjj(H) and so pintjt(G) n pinjH) = /. Consequently, Y is also prebinormal. □

Theorem 4.2. If f:(X, xj, x2) fi (Y, <rj, a2) is ij-pre-rgp-closed ij-R-map, surjection and X is mildly prebinormal then Y is also mildly prebinormal.

Proof. Let A 2 ij-RC(Y), B 2 ji-RC(Y) and A n B = /, Since f is surjection j-R-map, then f^1(A) 2 ij-RC(X), f^1(B) 2 ji-RC(X) and f~1(A) n f\B) = /. Since X is mildly prebinormal, then there exist U 2 ji-PO(X), V 2 ij-PO(X) and U n V = / such that /^1(A) ç U and f^1(B) ç V. Since f is ij-pre-rgp-closed, by Lemma 2.1, there exist G 2 ij-GPRO(Y) and H 2 ji-GPRO(Y) such that A ç G, B ç H, f^1(G) ç U and r\H) ç V. Since U and V are disjoint, G and H are disjoint. Since G 2 ij-GPRO(Y) and H 2 ji-GPO(Y), by Lemma 1.1 (4), then we have A ç pintj(G), B ç pintj(H) and so pintjt(G) n pintjj(H) = /.Consequently, Y is also mildly prebinormal. □

Theorem 4.3. If f:(X, sj, s2) fi (Y, <rj, a2) is ij-pre-rgp-closed ij-R-map, i-continuous, surjection and X is almost prebinormal, then Y is also almost prebinormal.

Proof. Let A 2 i-C(Y), B 2 ji-RC(Y) and A n B = /. Since f is ij-R-map, then, f^1(B) 2 ji-RC(X). Since f is i-continuous, then r\A) 2 i-C(X) and we have f\A) n f\B) = /. Since X is almost prebinormal, then there exist U 2 ji-PO(X), V 2 ij-PO(X) and U n V = / such that r\A) ç U and f^1(B) ç V. Since f is ij-pre-rgp-closed, by Lemma 2.1, there exist G 2 ij-GPRO(Y) and H 2 ji-GPRO(Y) such that A ç G, B ç H, fT1(G) ç U and f^1(H) ç V. Since U and V are disjoint, G and H are disjoint. Since G 2 ij-GPRO(Y) and H 2 ji-GPO(Y), by Lemma 1.1 (4), then we have A ç pintj(G), B ç pintj(H) and so pintji(G) n pintjj(H) = /. Consequently, Y is also almost prebinormal. □

Theorem 4.4. Iff:(X, xj, x2) fi (Y, <rj, a2) is ij-pre-gpr-continu-ous i-closed, injection and Y is prebinormal, then X is also prebinormal.

Proof. Let A 2 i-C(X), B 2 j-C(X) and A n B = /. Since f is i-closed injection, then f(A) 2 i-C(Y), f(B) 2 j-C(Y) and f(A) n f(B) = /.By prebinormality of Y, there exist U 2 ji-PO(Y), V 2 ij-PO(Y) and U n V = / such that f(A) ç U and f(B) ç V. Since f is j-pre-gpr-continuous,f^1(U) 2 ij-GPRO(X) and Г\Г) 2 ji-GPRO(X) such that A ç f\U), B ç /^1(f) and r\U) n f\V) = /. By Theorem 2.1(b), therefore, X is prebinormal. □

Theorem 4.5. Iff:(X, xj,x2) fi (Y, aj, a2) is ij-pre-gpr-continu-ous ij-rc-preserving, injection and Y is mildly prebinormal then Xis also mildly prebinormal.

Proof. Let A 2 ij-RC(X), B 2 ji-RC(X) and A n B = Since f is j-rc-preserving injection, then f(A) 2 ij-RC(Y), f(B) 2 ji-RC(Y) and f(A) nf(B) = /. By mild prebinormality of Y, there exist U 2 ji-PO(Y), V 2 ij-PO(Y) and U n V = / such that f(A) ç U and f(B) ç V. Since f is y'-pre-gpr-continuous, r\U) 2 ij-GPRO(X) and f\V) 2 ji-GPRO(X) such that A ç f\U), B ç r\V) and jTl(U) n Г\V) = /. By Theorem 2.3(c), therefore, X is mildly prebinormal. □

Theorem 4.6. Iff:(X,sj,s2) fi (Y,oj,o2) is ij-pre-gpr-continu-ous ij-rc-preserving, i-closed injection and Y is almost prebinormal then X is also almost prebinormal.

Proof. Let A 2 i-C(X), B 2 ji-RC(X) and A n B = /. Since f is ij-rc-preserving and i-closed injection, then f(A) 2 i-C(Y), f(B) 2 ji-RC(Y) andf(A) nf(B) = /. By almost prebinormality of Y, there exist U 2 ji-PO(Y), V 2 ij-PO(Y) and U n V = / such that f(A) ç U and f(B) ç V. Since f is ij-pre-gpr-continu-ous, f\U) 2 ij-GPRO(X) and f\V) 2 ji-GPRO(X) such that A çr\U), B çГ\У) andr\U) nГ\V) = /. By Theorem 2.2 (c), therefore, X is almost prebinormal. □

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