Journal of the Egyptian Mathematical Society (2015) 23, 527-534

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

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ORIGINAL ARTICLE

Generalized W -closed sets in bitopological spaces c^ark

H.M. Abu Donia a'*, M.A. Abd Allah b, A.S. Nawar b

a Department of Mathematics, Faculty of Science, Zagazig University, Egypt b Department of Mathematics, Faculty of Science, Minoufia University, Egypt

Received 25 January 2014; revised 4 December 2014; accepted 18 December 2014 Available online 2 February 2015

KEYWORDS

y-W'-closed sets;

ij-W -continuous functions;

ij — T1=5 spaces;

ij — T1W=5 spaces;

ij — w t1=5 spaces

Abstract In this paper, we introduce and study a new class of sets in a bitopological space (X, s1, s2), namely, ij-j -closed sets, which settled properly in between the class of ji-a-closed sets and the class of j-ga-closed sets. We also introduce and study new classes of spaces, namely, ij — T1=5 spaces, ij-Te spaces, ij-aTe spaces, ij-Tt spaces and ij-aTi spaces. As applications of ij-j -closed sets, we introduce and study four new classes of spaces, namely, ij — Tj^ spaces, ij — w T1=5 spaces (both classes contain the class of ij — T1=5 spaces), ij-aTk spaces and ij-Tk spaces. The class of ij-Tk spaces is properly placed in between the class of ij-Te spaces and the class of ij-Tt spaces. It is shown that dual of the class of ij — spaces to the class of ij-aTe spaces is the class of ij-aTk spaces and the dual of the class of ij — T1=5 spaces to the class of ij — T1=5 spaces is the class of ij — Tj=5 spaces and also that the dual of the class of ij-Tt spaces to the class of ij-Tk spaces is the class of ij-aTk spaces. Further we introduce and study ij-j -continuous functions and ij-j -irresolute functions.

2010 MATHEMATICAL SUBJECT CLASSIFICATION: 54 C 55; 54 C 10; 54 C 10; 54 E 55

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1. Introduction

Recently the topological structure s on a set X has a lot of applications in many real life applications. The abstractness of a set X enlarges the range of its applications. For example, a special type of this structure is the basic structure for rough set theory [1]. Alexandroff topologies are widely applied in the field of digital topologies [2]. Moreover, s and its generalizations are applied in biochemical studies [3].

The work presented in this paper will open the way for using two viewpoints in these applications. That is, to apply two topologies at the same time. The concepts of g-closed sets, gs-closed sets, sg-closed sets, ga-closed sets, ag-closed sets, gp-closed sets, gsp-closed sets and spg-closed sets have been introduced in topological spaces (cf. [4-10]). El-Tantawy and Abu-Donia [11] introduced the concepts of (ij-GC(X), ij-GSC(X), ij-SGC(X), ij-GaC(X), ij-aGC(X), ij-GPC(X), ij-GSPC(X), and ij-SPGC(X)) subset of (X, si, s2). Abd Allah and Nawar [12] introduced The concept of W -open sets and studied The properties of T1=5, Te, aTe, Tt, aT. In this paper, we introduce a new class of sets in a bitopological space (X, s1, s2), namely, ij-W -closed sets, which settled properly in between the class of ji-a-closed sets and the class of ij-ga-closed sets. And we extend the properties to a bitopological space (X, s1, s2). Also we use

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the family of ij-W'-closed sets to introduce some types of properties in (X, ti, t2), and we study the relation between these properties. The concepts of pre-continuous, semi-continuous, a-continuous, sp-continuous, g-continuous, ag-continuous, ga-continuous, gs-continuous, sg-continuous, gsp-continuous, spg-continuous, gp-continuous, gc-irresolute, gs-irresolute, ag-irresolute and ga-irresolute functions have been introduced in topological spaces (cf. [7,10,13-22]). El-Tantawy and Abu-Donia [11] introduced the concepts of (ij-pre-continuous, ij-semi-continuous, /-a-continuous, ij-sp-continuous, ij-g-contin-uous, /-ag-continuous, ij-ga-continuous, ij-gs-continuous, ij-sg-continuous, ij-gsp-continuous, ij-spg-continuous, ij-gp-con-tinuous, ij-gc-irresolute, ij-gs-irresolute, /-ag-irresolute and ij-ga-irresolute) functions in bitopological spaces. In this paper, we introduce a new functions in a bitopological space (X, t1, t2), namely, ij-\p -continuous functions and ij-\p -irresolute functions.

2. Preliminaries

Definition 2.1. [23] A subset A of a bitopological space (X, t1, t2) is called:

(1) ij-preopen if A c Trint(T,-cl(A)) and ij-preclosed if sr cl(s/-int(A)) c A.

(2) /-semi-open if A c t,-c1(t¿-int(A)) and ij-semi-closed if / int(ti-cl(A)) c A.

(3) ij-a-open if A c Trint("ij-cl(Trint(A))) and i/-a-closed if sr cl(Tj-int(Ti-cl(A))) c A.

(4) ij-semi-preopen if A c Tj-cl(Trint("r,-cl(A))) and ij-semi preclosed if tj-int(si-cl(sj-int(A))) c A.

The class of all ij-preopen (resp. /-semi-open, ij-a-open and ij-semi-preopen) sets in a bitopological space (X, t1, t2) is denoted by ij-PO(X) (resp. ij-SO(X), ij-aO(X) and ij-SPO(X)). The class of all ij-preclosed (resp. ij-semi-closed, ij-a-closed and ij-semi-preclosed) sets in a bitopological space (X, t1, t2) is denoted by ij-PC(X) (resp. ij-SC(X), ij-aC(X) and ij-SPC(X)).

Definition 2.2. [23] For a subset A of a bitopological space (X, t1, t2), the ij-pre-closure (resp. ij-semi-closure, ij-a-closure and ij-semi-pre-closure) of A are denoted and defined as follow:

(1) ij-pcl(A) = n(Fc X: F 2 ij-PC(X), F□ A}.

(2) ij-scl(A) = n(Fc X: F 2 ij-SC(X), F□ A}.

(3) ij-acl(A) = n(Fc X: F 2 ij-aC(X), F□ a}.

(4) ij-spcl(A) = n(Fc X: F 2 ij-SPC(X), F□ A}.

Dually, the ij-preinterior (resp. ij-semi-interior, ij-a-interior and ij-semi-preinterior) of A, denoted by ij-pint(A) (resp. ij-sint(A), ij-aint(A) and ij-spint(A)) is the union of all ij-preopen (resp. ij-semi-open, ij-a-open and ij-semi-preopen) subsets of X contained in A.

Definition 2.3. [11] A subset A of a bitopological space (X, t1, t2) is called:

(1) ij-g-closed (denoted by ij-GC(X)) if, A c U, U 2 st ) j-cl(A) c U.

(2) ij-gs-closed (denoted by ij-GSC(X)) if, A c U, U 2 -) ji-scl(A) c U.

(3) ij-sg-closed (denoted by ij-SGC(X)) if, A c U, U 2 ij-SO(X) ) ji-scl(A) c U.

(4) ij-ga-closed (denoted by ij-GaC(X)) if, A c U, U 2 ij-aO(X) ) ji-acl(A) c U.

(5) ij-ag-closed (denoted by ij-aGC(X)) if, A c U, U 2 -) ji-acl(A) c U.

(6) ij-gp-closed (denoted by ij-GPC(X)) if, A c U, U 2 -) ji-pcl(A) c U.

(7) ij-gsp-closed (denoted by ij-GSPC(X)) if, A c U, U 2 -) ji-spcl(A) c U.

(8) ij-spg-closed (denoted by ij-SPGC(X)) if, A c U, U2 ji-SPO(X)) ) ji-spcl(A) c U.

The complement of an ij-GC(X) (resp. ij-GSC(X), ij-SGC(X), ij-GaC(X), ij-aGC(X), ij-GPC(X), ij-GSPC(X), and ij-SPGC(X)) subset of (X, t1, t2) is called an ij-GO(X) (resp. ij-GSO(X), ij-SGO(X), ij-GaO(X), ij-aGO(X), ij-GPO(X), ij-GSPO(X), and ij-SPGO(X)) subset of (X, t1, t2).

Definition 2.4. [11] A function f: (X,

t1, S2) fi (Y, ^1, ff2) is

called:

(1) ij-pre-continuous if " V 2 i-C(Y), f ^(F) 2 ij-PC(X).

(2) ij-semi-continuous if " V2 i-C(Y), f ~l(V) 2 ij-SC(X).

(3) ij-a-continuous if " V2 i-C(Y), f ~l(V) 2 ij-aC(X).

(4) ij-sp-continuous if " V 2 i-C(Y), f ^(V) 2 ij-SPC(X).

(5) ij-g-continuous if " V 2 j-C(Y), f ^(V) 2 ij-GC(X).

(6) ij-ag-continuous if " V 2 j-C(Y), f _1(V) 2 ij-aGC(X).

(7) ij-ga-continuous if V V2 j-C(Y), f X(V) 2 ij-GaC(X).

(8) ij-gs-continuous if V V 2 j-C(Y), f ^(V) 2 ij-GSC(X).

(9) ij-sg-continuous if V V 2 j-C(Y), f ^(v) 2 ij-SGC(X).

(10) ij-gsp-continuous if V V 2 j-C(Y), f~l (V) 2 ij-GSPC(X).

(11) ij-spg-continuous if V V 2 j-C(Y), f ^(v) 2 ij-SPGC(X).

(12) ij-gp-continuous if V V 2 j-C(Y), f ^(V) 2 ij-GPC(X).

(13) i-continuous if V V 2 i-C(Y), f ^(V) 2 i-C(X).

(14) ij-gc-irresolute if V V 2 ij-GC(Y), f^(V) 2 ij-GC(X).

(15) ij-gs-irresolute if V V 2 ij-GSC(Y), f ^(V) 2 ij-GSC(X).

(16) ij-ag-irresolute if V V 2 ij-aGC(Y), fl(V) 2 ij-aGC(X).

(17) ij-ga-irresolute if V V 2 ij-GaC(Y), fl(V) 2 ij-GaC(X).

Definition 2.5. [12] A subset A of (X, t) is called W*-closed if A c U, U 2 GaO(X) ) acl(A) c U. The complement of W*-closed set is said to be W -open.

Definition 2.6. [12] A space (X, t) is called:

(1) T1/5 space if GaC(X) = aC(X).

(2) TW/5 space if W*C(X) = a.C(X).

(3) W*:T1/5 space if GaC(X) = W*C(X).

(4) Te space if GSC(X) = aC(X).

(5) aTe space if aGC(X) = aC(X).

(6) Tk space if GSC(X) = W*C(X).

(7) aTk space if aGC(X) = W*C(X).

(8) Ti space if GSC(X) = GaC(X).

(9) aTl space if aGC(X) = GaC(X).

Definition 2.7. [12] A function f: (X, t) fi (Y, r) is called:

(1) ^'-continuous if V V 2 C(Y), f ^(V) 2 W*C(X).

(2) W'-irresolute if V V 2 W*C(Y), f ^1(V) 2 W*C(X).

(3) pre-W*-closed if A 2 W*C(X), f (A) 2 W*C(Y).

3. Basic properties of ij-ф -closed sets

Proof.

We introduce the following definition.

Definition 3.1. A subset A of a bitopological space (X, ть t2) is called ij-ф -closed set if, A с U, U 2 ji-GaO(X) ) ji-acl(A) с U.

The class of y'-W'-closed subsets of (X, ть s2) is denoted by

ij-W*c(X).

The following diagram shows the relationships of ij-ф -closed sets with some other sets discussed in this section (see Diagram 1).

Definition 3.1 is a particular case of Definition 8 from Noiri [24].

Theorem 3.1. Every ji-a-closed set is an ij-ф -closed set.

The following example supports that an y'-W'-closed set need not be a ji-a-closed set in general.

Example 3.1. Let X = {a, b, c, d}, t1 = {X, ф, {a}, {a, d}} and t2 = {X, ф, {a, b}, {c, d}}. Then we have A = {b, c} 2 ij-W*C(X) but A R ji-a.C(X).

Therefore the class of y'-W'-closed sets is properly contains the class of ji-a-closed sets. Next we show that the class of ij-ф -closed sets is properly contained in the class of y'-ga-closed set.

Theorem 3.2. Every ij-W*-closed set is an ij-ga-closed set.

The following example supports that the converse of the above theorem is not true in general.

(1) Let A 2 ij-ф C(X) n ji-GaO(X). Then we have ji-acl(A) ç A. Consequently, A 2 ji-aC(X).

(2) Let U 2 ji-GaO(X) such that B ç U. Since A ç B and A 2 ij-ф C(X), then ji-acl(A) ç U. Since B ç ji-acl(A), then we have ji-acl(B) ç ji-acl(A) ç U. Therefore, B 2 ij-W*C(X). □

Theorem 3.4. Let (X, xj, x2) be a bitopological space, A 2 ij-GaC(X). Then A 2 ij-W*C(X) ifij-aO(X) = ji-GaO(X).

Proof. Let A 2 ij-GaC(X) i.e. A ç U and U 2 ij-aO(X), then ji-acl(A) ç U. Since ij-aO(X) = ji-GaO(X). Consequently, A ç U and U 2 ji-GaO(X), then ji-acl(A) ç U i.e. A 2 ij-W*C(X). □

Theorem 3.5. Let (X1, s1, x2) and (X2, x*, s2) be two bitopological spaces. Then the following statement is true. If A 2 ij-W*O(X1) and B 2 ij-W*O(X2), then A x B 2 ij-W*O(X1 x X2).

Proof. Let A 2 ij-ф*OX) and B 2 ij-W*O(X2) and W = A x B ç X1 x X2. Let F = F1 x F2 ç W, F 2 ji-GaC(X1 x X2). Then there are F1 2 ji-Ga.C(X1), F2 2 ji-Ga.C(X2), F1 ç A, F2 ç B and so, F1 # tji — aint(A) and F2 С — aint(B). Hence F1 x F2 С A x BandF1 x F2 Сxß — aint(A)x x^ — aint(B) — Xji x t*. — aint(A x B).

Therefore A x B 2 ij-ф*OX x X2). □

Theorem 3.6. A subset A of X is ij-ф O(X) if and only if F is a subset of ij-aint(A) whenever F ç A and F 2 ji-GaC(X).

Example 3.2. Let X, s1, and s2 are as in the Example 3.1. Then the subset B = {b} 2 ij-GaC(X) but B R ij-j*C(X).

Theorem 3.7. For each x 2 X, either {x} is ji-GaC(X) or {x} is ij-ф*O(X).

Remark 3.1. The intersection of two sets in ij-j -closed set is not in general a set in ij-j -closed set, as shown by the following example.

Example 3.3. Let X, s1, and s2 be as in the Example 3.1. Then we have {a, b} and {b, c} 2 ij-j*C(X) but {a, b} n {b, c} = {b} R ij-W*C(X).

Theorem 3.3. For any bitopological space (X, sj, s2).

(1) ij-W*C(X) n ji-GaO(X) c ji-aC(X).

(2) If A 2 ij-j C(X) and A c B c ji-acl(A), then B 2 ij-W*C(X).

Theorem 3.8. A subset A of X is ij-ф C(X) if and only if ji-aC(A) n F = 0, whenever A n F = 0, where F is ji-GaC(X).

4. Applications of ij-ф -closed sets

As applications of i/^-closed sets, four new classes of spaces, namely, ij — Тф*5 spaces, ij — ф* T1=5 spaces, ij-Tk spaces and ij-aTk spaces are introduced.

We introduce the following definitions.

Definition 4.1. A bitopological space (X, x1, x2) is called an ij — T1=5 space if ij-GaC(X) = ji-aC(X).

j-closed -

ji- a-closed •

ij- g-closed

->■ ij- ^""-closed

ji- semi-closed -► ij- sg-clos

ji- semi-preclosed

ij- ag-closed

ij-gp-closed

■>■ ij- gsp-closed

ji-preclosed

Diagram 1

Definition 4.2. A bitopological space (X, t1, t2) is called an ij - T/5 space if ij-W*C(X) = ji-aC(X).

We prove that the class of ij — T^ spaces properly contains the class of ij - T1/5 spaces.

Theorem 4.1. Every ij — T1/5 space is an ij — T^ space.

Proof. Follows from the fact that every i/-W*-closed set is an ij-ga-closed set. □

The converse of the above theorem is not true as it can be seen from the following example.

Example 4.1. Let X = (a, b, c}, t1 = (X, /, (a}} and t2 = (X, /, (b}}. Then (X, t1, t2) is an ij — TW/5 space but not an ij — T1/5 space since (b, c} 2 ij-GaC(X) but (b, c} R ji-aC(X).

We introduce the following definition.

Definition 4.3. A bitopological space (X, t1, t2) is called an ij — W T1/5 space if ij-GaC(X) = ij-W*C(X).

Theorem 4.2. Every ij — T1/5 space is an ij — w T1/5 space.

Proof. Let (X, t1, t2) be an ij — T1/5 space. Let A 2 ij-GaC(X). Since (X, t1, t2) is an ij — T1/5 space, then A 2 ji-aC(X). Hence, by using Theorem 3.1, we have A 2 ij-W C(X). Therefore (X, t1, t2) is an ij — W T1/5 space. □

The converse of the above theorem is not true as we see in the following example.

Example 4.2. Let X = (a, b, c}, t1 = (X, /, (a}} and t2 = (X, /, (a}, (b, c}}. Then (X, t1, t2) is an ij — w T1/5 space but not an ij — T1/5 space since (a, b} 2 ij-GaC(X) but (a, b} R ji-

aC(X).

We show that ij — T^ ness is independent from ij — T1/5 ness.

Remark 4.1. ij — TW/5 ness and ij — T1/5 ness are independent as it can be seen from the next two examples.

Example 4.3. Let X, t1, and t2 be as in the Example 4.1. Then (X, t1, t2) is an ij — T^ space but not an ij — W T1/5 space since (b, c} 2 ij-GaC(X) bu (b, c} R ij-W*C(X).

Example 4.4. Let X, t1, and t2 be as in the Example 4.2. Then (X, t1, t2) is an ij — W T1/5 space but not an ij — T^ space since (a, c} 2 ij-W*C(X) but (a, c} R ji-aC(X).

Theorem 4.3. If (X, tj, t2) is an ij — w T1/5 space, then for each x 2 X, {x} is either ji-a-closed or ij-W -open.

Proof. Suppose that (X, t1, t2) is an ij — T1/5 space. Let x 2 X and assume that (x} R ji-aC(X). Then (x} R ij-GaC(X) since every ji-a-closed set is an ij-ga-closed set. So X-(x} R ji-aO(X). Therefore X-(x} 2 ij-GaC(X) since X is the only ji-a-open set which contains X-(x}. Since (X, t1, t2) is an ij —W* t1/5 space, then X-(x} 2 ij-W C(X) or equivalently (x} 2 ij-W*O(X). □

Theorem 4.4. A space (X, tj, t2) is an ij — T1/5 space if and only if it is ij — T1/5 and ij — T'W/5 space.

Proof. The necessity follows from the Theorems 4.1 and 4.2. For the sufficiency, suppose that (X, t1, t2) is both ij — T1/5 and ij — T^ space. Let A 2 ij-GaC(X). Since (X, t1, t2) is an ij —W* t1/5 space, then A 2 ij-W C(X). Since (X, t1, t2) is an ij — TW/5 space, then A 2 ji-aC(X). Thus (X, t1, t2) is an ij — T1/5 space. □

We introduce the following definitions ij-Te spaces and ij-aTe spaces respectively and show that every ij-Te (ij-aTe) space is an ij — T1/5 space.

Definition 4.4. A space (X, t1, t2) is called an ij-Te space if ij-GSC(X) = ji-aC(X).

Definition 4.5. A space (X, t1, t2) is called an ij-aTe space if ij-aGC(X) = ji-aC(X).

Theorem 4.5. Every ij-Te space is an ij — T1/5 space.

Proof. Follows from the fact that every ij-ga-closed set is an ij-gs-closed set. □

An ij — T1/5 space need not be an ij-Te space as we see the next example.

Example 4.5. Let X = (a, b, c}, t1 = (X, /, (a}, (b}, (a, b}, (a, c}} and t2 = (X, /, (a}, (a, b}}. Then (X, t1, t2) is an ij — T1/5 space but not an ij-Te space since (b} 2 ij-GSC(X) but (b} R ji-aC(X).

Theorem 4.6. Every ij-aTe space is an ij — T1/5 space.

Proof. Follows from the fact that every ij-ga-closed set is an ij-ag-closed set. □

An ij — T1/5 space need not be an ij-aTe space as we see the next example.

Example 4.6. Let X = (a, b, c}, t1 = (X, /, (a}, (b}, (a, b}} and t2 = (X, /, (a}, a, c}}. Then (X, t1, t2) is an ij — T1/5 space but not an ij-aTe space since (a, c} 2 ij-aGC(X) but (a, c} R ji-aC(X).

Theorem 4.7. Every ij-Te space is an ij-aTe space.

Proof. Follows from the fact that every ij-ag-closed set is an ij-gs-closed set. □

The converse of the above theorem is not true in general as the following example supports.

Example 4.7. Let X, t1, and t2 be as in the Example 4.5. Then (X, t1, t2) is an ij-aTe space but not an ij-Te space since (b} 2 ij-GSC(X) but (b} R ji-aC(X).

Theorem 4.8. Every ij-Te space is an ij — T^ space.

Proof. Follows from the fact that every ij-W*-closed set is an ij-gs-closed set. □

The converse of the above theorem is not true in general as the following example supports.

Example 4.8. Let X = {a, b, c, d, e}, s1 = {X, /, {a}, {c, d}, {a, c, d}, {b, c, d, e}} and s2 = {X, /, {a}, {a, b}, {a, b, e}, {a, c, d}, {a, b, c, d}}. Then (X, s1, s2) is an ij — Tj=5 space but not an ij-Te space since {d} 2 ij-GSC(X) but {d} R ji-aC(X).

Theorem 4.9. Every ij-aTe space is an ij — Tj=5 space.

Proof. Follows from the fact that every i/'-j'-closed set is an i/-ag-closed set. □

An ij — T(=5 space need not be an ij-aTe space as we see the next example.

Example 4.9. Let X, s^ and s2 be as in the Example 4.8. Then (X, s1, s2) is an ij — Tj,5 space but not an ij-aTe space {c} 2 ij-aGC(X) but {c} R ji-a£5X).

We introduce the following definitions.

Definition 4.6. A space (X, s1, s2) is called an ij-Tk space if ij-GSC(X) = ij-fC(X).

Definition 4.7. A space (X, s1, s2) is called an ij-aTk space if ij-aGC(X) = ij-fC(X)

The following example supports that the converse of the above theorem is not true in general.

Example 4.11. Let X, s1, and s2 be as in the Example 4.1. Then (X, s1, s2) is an ij-aTi space but not an ij-aTk space since {b} 2 ij-aGC(X) but {b} R ij-fC(X).

Theorem 4.12. A space (X, sj, s2) is an ij-aTe space if and only if it is ij-aTk and ij — space.

Proof. The necessity follows from the Theorems 4.9 and 4.10. For the sufficiency, suppose that (X, s1, s2) is both ij-aTk and ij — T/5 space. Let A 2 ij-aGC(X). Since (X, s1, s2) is an ij-aTk space, then A 2 ij-\j/* C(X). Since (X, s1, s2) is an ij — Tj/5 space, then A 2 ji-aC(X). Thus (X, s1, s2) is an ij-aTe space. □

Remark 4.2. ij-a.Tk ness and ij — Tj/5 ness are independent as it can be seen from the next two examples.

Example 4.12. Let X, s1, and s2 be as in the Example 4.2. Then (X, s1, s2) is an ij-a.Tk space but not an ij — Tj/5 space since {a, b} 2 ij-j*C(X) but {a, b} R ji-aC(X).

Example 4.13. Let X, s1, and s2 be as in the Example 4.1. Then (X, s1, s2) is an ij — T/5 space but not an ij-a.Tk space since {b, c} 2 ij-aGC(X) but {b, c} R ij-fC(X).

Definition 4.8. A space (X, s1, s2) is called an ij-Ti space if ij-GSC(X) = ij-GaC(X).

Definition 4.9. A space (X, s1, s2) is called an ij-aTl space if ij-aGC(X) = ij-GaC(X).

We show that the class of ij-aTk spaces properly contains the class of ij-aTe spaces and is properly contained in the class of ij-aTl spaces. We also show that the class of ij-aTk spaces is the dual of the class of ij — Tj/5 spaces to the class of ij-aTe spaces. Moreover we prove that ij-a.Tk ness and ij — Tj/5 ness are independent from each other.

Theorem 4.10. Every ij-aTe space is an ij-aTk space.

Proof. Let (X, s1, s2) be an ij-aTe space. Let A 2 ij-aGC(X). Since (X, s1, s2) is an ij-aTe space, then A 2 ji-aC(X). Hence, by using Theorem 3.1, we have A 2 ij-j C(X). Therefore (X, s1, s2) is an ij-aTk space. □

The following example supports that the converse of the above theorem is not true in general.

Example 4.10. Let X, s1, and s2 be as in the Example 4.2. Then (X, s1, s2) is an ij-aTk space but not an ij-aTe space since {a, c} 2 ij-aGC(X) but {a, c} R ji-aC(X).

Theorem 4.11. Every ij-aTk space is an ij-aTl space.

Proof. Let (X, s1, s2) be an ij-aTk space. Let A 2 ij-aGC(X). Since (X, s1, s2) is an ij-aTk space, then A 2 ij-j C(X). Hence, by using Theorem 3.2, we have A 2 ij-GaC(X). Therefore (X, s1, s2) is an ij-aTl space. □

Definition 4.10. A subset A of a bitopological space (X, s1, s2) is called an ij-j*-open if its complement is an ij-j*-closed of (X, s1, s2).

Theorem 4.13. If (X, sj, s2) is an ij-aTk space, then for each x 2 X, {x} is either ij-ag-closed or ij-j -open.

Proof. Suppose that (X, s1, s2) is an ij-aTk space. Let x 2 X and assume that {x} R ij-aGC(X). Then {x} R ji-aC(X) since every ji-a-closed set is an i/-ag-closed set. So X-{x} R ji-aO(X). Therefore X-{x} 2 ij-aGC(X) since X is the only ji-a-open set which contains X-{x}. Since (X, s1, s2) is an ij-aTk space, then X-{x} 2 ij-j*C(X) or equivalently {x} 2 ij-

w*o(x). □

Theorem 4.14. Every ij-aTk space is an ij — w T1/5 space.

Proof. Let (X, s1, s2) be an ij-aTk space. Let A 2 ij-GaC(X), then A 2 ij-aGC(X). Since (X, s1, s2) is an ij-aTk space, then A 2 ij-j C(X). Therefore (X, s1, s2) is an ij — w T1/5 space. □

The following example supports that the converse of the above theorem is not true in general.

Example 4.14. Let X, s1, and s2 be as in the Example 4.8. Then (X, s1, s2) is an ij — w T1/5 space but not an ij-aTk space since {c} 2 ij-aGC(X) but {c} R ij-fC(X).

We show that the class of ij-Tk spaces properly contains the class of ij-Te spaces, and is properly contained in the class of ij-aTk spaces, the class of ij-Tl spaces, and the class of ij-aTi spaces.

Theorem 4.15. Every ij-Te space is an ij-Tk space.

Proof. Let (X, ti, t2) be an ij-Te space. Let A 2 ij-GSC(X). Since (X, ti, t2) is an ij-Te space, then A 2 ji-aC(X). Hence, by using Theorem 3.1, we have A 2 ij-W C(X). Therefore (X, t1, t2) is an ij-Tk space. □

The following example supports that the converse of the above theorem is not true in general.

Example 4.15. Let X, t1, and t2 be as in the Example 4.2. Then (X, t1, t2) is an ij-Tk space but not an ij-Te space since {a, c} 2 ij-GSC(X) but {a, c} R ji-aC(X).

Theorem 4.16. Every ij-Tk space is an ij-aTk space.

Proof. Let (X, t1, t2) be an ij-Tk space. Let A 2 ij-aGC(X), then A 2 ij-GSC(X). Since (X, t1, t2) is an ij-Tk space, then A 2 ij-W C(X). Therefore (X, t1, t2) is an ij-aTk space. □

The converse of the above theorem is not true as it can be seen from the following example.

Example 4.16. Let X, t1, and t2 be as in the Example 4.5. Then (X, t1, t2) is an ij-aTk space but not an ij-Tk space since {b} 2 ij-GSC(X) but {b} R ij-W * C(X).

ij " Ti

ij - Ti

ij-aTe

+ij -' T

ij-aTk

Diagram 2

-> ij-oTi

The following diagram shows the relationships between the separation axioms discussed in this section (see Diagram 2).

5. i/-W*-continuous and j-W*-irresolute functions

We introduce the following definition.

Theorem 4.17. Every ij-Tk space is an ij-Tl space.

Proof. Let (X, t1, t2) be an ij-Tk space. Let A 2 ij-GSC(X). Since (X, t1, t2) is an ij-Tk space, then A 2 ij-W C(X). Hence, by using Theorem 3.2, we have A 2 ij-GaC(X). Therefore (X, t1, t2) is an ij-Ti space. □

The converse of the above theorem is not true as it can be seen from the following example.

Example 4.17. Let X = {a, b, c}, t1 = {X, /, {a, b}} and t2 = {X, /, {a, c}}. Then (X, t1, t2) is an ij-Tt space but not an ij-Tk space since {c} 2 ij-GSC(X) but {c} R ij-W * C(X).

Next we prove that the dual of the class of ij-Ti spaces to the class of ij-Tk spaces is the class of ij-aTk spaces.

Theorem 4.18. A space (X, tj, t2) is an ij-Tk space if and only if it is ij-aTk and ij-Tl space.

Proof. The necessity follows from the Theorems 4.16 and 4.17. For the sufficiency, suppose that (X, t1, t2) is both ij-aTk and ij-Tt space. Let A 2 ij-GSC(X). Since (X, t1, t2) is an ij-Tl space, then A 2 ij-GaC(X). Then A 2 ij-aGC(X). Since (X, t1, t2) is an ij-aTk space, then A 2 ij-W C(X). Therefore (X, t1, t2) is an ij-Tk space. □

Theorem 4.19. A space (X, tj, t2) is an ij-Te space if and only if it is ij-Tk and ij — space.

Proof. The necessity follows from the Theorems 4.8 and 4.15. For the sufficiency, suppose that (X, t1, t2) is both ij-Tk and ij — rW/5 space. Let A 2 ij-GSC(X). Since (X, t1, t2) is an ij-Tk space, then A 2 ij-W C(X). Since (X, t1, t2) is an ij — TW/5 space, then A 2 ji-aC(X). Therefore (X, t1, t2) is an ij-Te space. □

Definition 5.1. A functionf: (X, t1, t2) fi (Y, r1, r2) is called ij-W*-continuous if " V2 j-C(Y), f—l[V) 2 ij-W*C(X).

The following diagram shows the relationships of ij-W*-con-tinuous functions with some other functions discussed in this section (see Diagram 3).

Theorem 5.1. Every ji-a-continuous function is ij-W -continuous.

The following example supports that the converse of the above theorem is not true in general.

Example 5.1. Let X = {a, b, c, d}, Y = {u, v, w}, t1 = {X, /, {a},{a, d}}, t2 = {X, /, {a, b},{c, d}}, n = { Y, /, {u}, {v}, {u, v}, {u, w}} and r2 = {Y, /, {u}, {u, v}}. Define f: (X, t1, t2) fi (Y, r1, r2) by f (a) = u, f (b) = v and f (c) = f (d) = w. f is not ji-a-continuous function since {v, w} 2 j-C(Y) but f 1({v, wg) = fb; c, dg Rji-aC(X). However f is i/-W*-continu-ous function.

Theorem 5.2. Every ij-W -continuous function is ij-ga-continuous.

The following example supports that the converse of the above theorem is not true in general.

Example 5.2. Let X, Y, t1, t2, r1 and r2 be as in the example 5.1. Define f: (X, t1, t2) fi (Y, r2) by f (a) = u, f (b) = w and f (c) = f (d) = v. f is not ij-W*-continuous function since {w} 2 j-C(Y) but f—1 ({wg) = {bg R ij-W*C(X). However f is ij-ga-continuous function.

Theorem 5.3. Iffj: (Xj, sj, s2) fi (Yj, rj, r2) andf2: (X2, sj, t2) fi (Y2, oj, r2) be two ij-W -continuous functions. Then the function f: (Xj X X2, sj X tj, t2 X t2) fi (Yj X Y2, rj X rj, 02 X o2) defined by f (xj, X2) = (f (xj), f (X2)) is ij-W*-continuous.

j-continuous-► ij-g- continuous->-ij- ag-continuous -► ij-gp-continuous

ji-a-continuous-'-continuous-► j-ga-continuous j-gs- continuous

< ' ^^^^^^ 1 r

ji-semi-continuous-► j-sg-continuous-► j-spg-continuous-► ij-g^rartmuo^

1' _____—""" ^ ^

ji- semi-precontinuous ^- ji-pre-continuous

Diagram 3

Proof. Let Vj 2 j-O(Y1) and V2 2 j-O(Y2). Sincef1 andf2 are two i/'-j'-continuous, then f—l(V1)2ij — j*O(X1) and f— (V2) 2 ij — W'O(X2). Hence, by using Theorem 3.5, we have

J—1(V1)xJ—1(V2)2ij — rO(X1 x X2). □

We introduce the following definition.

Definition 5.2. A functionf: (X, s1, s2) fi (Y, r1, o2) is called ij-j'-irresolute if " V 2 ij-j*C(Y), f—1(V) 2 ij-j*C(X).

Theorem 5.4. Every ij-j -irresolutefunction is ij-j -continuous.

The following example supports that the converse of the above theorem is not true in general.

Example 5.3. Let X = {a, b, c, d}, Y = {u, v, w}, s1 = {X, /, {a}, {a, d}}, s2 = {X, /, {a, b}, {c, d}}, o1 = {Y, /, {u}} and o2 = { Y, /, {u}, {v, w}}. Definef: (X, s1, s2) fi (Y, r1, o2) by f (a) = v, f (b) = wand f (c) = f (d) = u. f is not ij-j -irresolute function since {u, v} 2 ij-W*C(Y) but f— 1({u, vg) — {a, c, dg R ij — Wi*C(X). Howeverf is ij-j -continuous function.

Theorem 5.5. Let f: (X, sj, s2) fi (Y, rj, o2) and g: (Y, rj, o2) fi (Z, gj, g2) be any two functions. Then

(1) g o f is ij-j *-continuous if g is j-continuous and f is ij-j * -continuous.

(2) g of is ij-j -irresolute if both f and g are ij-j -irresolute.

(3) g of is ij-j -continuous if g is ij-j -continuous and f is ij-W -irresolute.

Proof. Let V 2 j-C(Z), since g is j-continuous, then g—1(V) 2 j-C(Y). Since f is ij-j -continuous, then we have f—l (g—1 (V)) 2 ij-W C(X). Consequently, g o f is ij-j -continuous.

(2)-(3) Similarly. □

Theorem 5.6. Let f: (X, sjt s2) fi (Y, rj, o2) be an ij-j*-continuous function. If (X, sjt s2) is ij — space, then f is ji-a-continuous function.

Proof. Let V 2j-C(Y). Since f is ij-j*-continuous, then f—1(V) 2 ij-j*C(X). Since (X, s1, s2) is an ij — Tj=5 space, then f—l(V) 2ji-aC(X). Consequently, f is ji-a-continuous. □

Theorem 5.7. Let f: (X, sj, s2) fi (Y, rj, o2) be an ij-ag-continuous function. If (X, sj, s2) is an ij-aTk space, then f is ij-w*-continuous.

Proof. Let V2 j-C(Y). Since f is an i/-ag-continuous function, thus f—1(V) 2 ij-aGC(X). Since (X, s1, s2) is an ij-aTk space, thenf—l(V) 2 ij-j C(X). Consequently, f is ij-j -continuous. □

Theorem 5.8. Let f: (X, sj, s2) fi (Y, rj, o2) be an ij-ga-con-tinuous function. If (X, sj, s2) is ij — W T1/5 space, then f is ij-j -continuous.

Proof. Let V2 j-C(Y). Since f is an i/-ga-continuous function, thusf—l (V) 2 ij-GaC(X). Since (X, s1, s2) is an ij — T1/5 space, thenf—l(V) 2 ij-j C(X). Consequently, f is ij-j -continuous. □

Theorem 5.9. Let f: (X, sj, s2) fi (Y, rj, o2) be an ij-gs-con-tinuous function. If (X, sjr s2) is ij-Tk space, then f is ij-j -continuous.

Proof. Let V2 j-C(Y). Since f is an i/-gs-continuous function, thus f—l(V) 2 ij-GSC(X). Since (X, s1, s2) is an ij-Tk space, then f—l (V) 2 ij-j*C(X). Consequently, f is ij-j'-continuous. □

Theorem 5.10. Let f: (X, sjt s2) fi (Y, rj, o2) be onto, ij-j*-irresolute and ji-a-closed. If (X, sj, s2) is ij — Tj^ space, then (Y, rj, o2) is also an ij — space.

Proof. Let V 2 ij-j*C(Y). Since f is ij-j*-irresolute, then f—l(V)2ij-W*C(X). Since (X, s1, s2) is ij — Tf/5 space, then f—l (V) 2 ji-aC(X). Since f is ji-a-closed and onto. Then we have V 2 ji-aC(Y). Therefore (Y, r1, o2) is also an ij — Tj/5 space. □

We introduce the following definition.

Definition 5.3. A functionf: (X, s1, s2) fi (Y, r1, o2) is called an i/-pre-W*-closed if A 2 ij-j*C(X), f(A) 2 ij-j*C(Y).

Theorem 5.11. Let f: (X, sj, s2) fi (Y, rj, o2) be onto, ij-ga-irresolute and ij-pre-j -closed. If (X, sj, s2) is ij — W T1/5 space, then (Y, rj, o2) is also an ij — W T1/5 space.

Proof. Let V2 ij-GaC(Y). Since f is i/-ga-irresolute, then f—l (V) 2 ij-GaC(Y). Since (X, s1, s2) is an ij — W T1/5 space. Since f is ij-pre-j -closed and onto. Then we have f(f—l (V)) — V 2 ij-W C(Y). Therefore (Y, r1, o2) is also an ij — W T1/5 space. □

Theorem 5.12. Let f: (X, sj, s2) fi (Y, rj, o2) be onto, ij-ag-irresolute and ij-pre-j -closed. If (X, sjr s2) is an ij-aTk space, then (Y, rj, o2) is also an ij-aTk space.

Proof. Let V 2 ij-aGC(Y). Since f is ij-ag-irresolute, then (V) 2 ij-aGC(X). Since (X, t1; t2) is an ij-aTk space, then fl (V) 2 ij-W C(X). Since f is i/-pre-W -closed and onto. Then we have f (V)) = V 2 ij-W * C( Y). Therefore (Y, r1, r2) is also an ij-aTk space. □

Theorem 5.13. Let f: (X, tj, t2) fi (Y, oJ, o2) be onto, ij-gs-irresolute and ij-pre-W -closed. If (X, tj, t2) is an ij-Tk space, then (Y, oJ, o2) is also an ij-Tk space.

Proof. Let V 2 ij-GSC(Y). Since f is ij-gs-irresolute, then fl (V) 2 ij-GSC(X). Since (X, t1, t2) is an ij-Tk space, then fl (V) 2 ij-W C(X). Since f is ij-pre-W -closed and onto. Then we have f(f1 (V)) = V 2 ij-W * C(Y). Therefore (Y, r1, r2) is also an ij-Tk space. □

Acknowledgments

The authors are greatly indebted to an anonymous referee for a very careful reading and pointing out necessary corrections.

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