ON WEAKLY ?-CONTINUOUS FUNCTIONS IN
BITOPOLOGICAL SPACES
Azal Jaafer Moosa Meera Nada Mohammed Abbas
Babylon University
Babylon University
College of Education College of Education Mathematics department Mathematics department
Samah Abd Al- hadi Babylon University
College of science for girls computer department
Abstract
. As a generalization of ? -continuous functions, we introduce and study several properties of weakly ? -continuous functions in Bitopological spaces and we obtain its several characterizations Keywords and phrases. Bitopological spaces, ? -open sets ,weakly ? -continuous function.
هصلاخلأ
فيزعتل ثحبلا يف ةمادختسا مت دق ]1[ يف هفيزعت مت يذلا يجولوبتلا يئانث ءاضف يف
عوضوم نا
? -continuous
ةلثملأا و تايزظنلا ضعب غم
1. Introduction
almost ?-continuous و weakly ? -continuous يهو ةيرازمتسلاا يف فعضا عاونا
The notion of ? -open sets due to al-talkany[1] , semi-preopen sets due to Andrijevi? [2] plays a significant role in general topology. In [3] the concept of ? -continuous functions is introduced and further Popa and Noiri[5] studied the concept of weakly ? -continuous functions. .In this paper,we introduce and study the notion of weakly ? -continuous functions in bitopological spaces further and investigate the properties of these functions.
Throughout the present paper,(X,T,T?)(resp.( X,?))) denotes a bitopological (resp.topological) space.Let (X,?) be a topological space and A be a subset of X.The closure and interior of A are denoted by Cl(A) and Int(A) respectively.
Let ,(X,T,T?) be a bitopological space and let A be a subset of X.The closure and interior of A with respect to T or T? are denoted by ClT(A) , intT(A)or ClT?(A) and IntT?(A),respectively.
2. basic definition
In this section we give all basic definition and some theorems and lemma we needs in this paper.
Definition 2.1 [1]. A subset A of a bitopological space ,(X,T,T?)) is said to be (i)regular open if A=IntT((ClT?(A)) .
(ii).regular closed if A=ClT ((IntT?(A)) .
(iii).preopen if A? IntT((ClT?(A)) . Remark 2.1:
1. ?-interior mean that the interior w.r.t. ?-open set.
2. ?-cl mean clouser w.r.t. ?-open set.
Definition 2.2.[1] A subset A of a bitopological space ,(X,T,T?) is said to be ? -open if there exist T?-open set U such that A?U , A?ClT(U).
Lemma 2.1.[1]Let (X,?1,?2) be a bitopological space and A be a subset of X.Then
(i).A is ?-open if and only if A=?Int(A).
(ii).A is ?-closed if and only if A=?Cl(A).
523Journal of Kerbala University , Vol. 10 No.3 Scientific . 2012
Lemma 2.2.For any subset A of a bitopological space ,(X,T,T?),x??Cl(A)if and only if U? A? ? for every ?-open set U containing x.
Definition 2.3.[4]A function f,(X,T,T?)? (Y,K,K?) is said to be ?-continuous if f-1(V) is ?-open in X for each K-open set V of Y.
3.Weakly ?- continuous
In this section we define weakly ? –continuous with some theorems
Definition 3.1.(i). A function f,(X,T,T?)? (Y,K,K?) is said to be weakly precontinuous if for each x?X and each K-open set V of Y containing f(x),there exists preopen set U containing x such that f(U)?ClT?(V).
(ii). A function f:(X,T,T?)? (Y,K,K?) is said to be weakly- ?-continuous if for each x?X and each K-open set V of Y containing f(x),there exists ?-open set U containing x such that f(U)?ClT?(V). A function f,(X,T,T?) ? (Y,K, K?) is said to be pairwise weakly precontinuous (resp.pairwise weakly ?-continuous) if f is weakly precontinuous and weakly -precontinuous (resp. if f is weakly ?-continuous )
Example 3.1 . Let X={a,b,c,d},T={ X, ?,{a}, {b,c} , {a,b,c}}, T=T? ? – open ( X) =0
Y={1,2,3},K={Y, ?,{1}},K?={X, ?,{1},{1,2},{2,3}}
let f:(X,T,T?)? (Y,K,K?) defined by f(a)=1,f(b)=f(c)=2 then f is weakly ?- continuous . Remark 3.1 :The composition of two weakly ?- continuous is not necessary weakly ?- continuous
Theorem 3.2.For a function f:(X,T,T?)? (Y,K,K?),the following properties are equivalent:
(i). f is weakly ?-continuous.
(ii). ?Cl(f-1 (IntT?(ClT(B))))) ?f-1 (ClT(B)) for every subset B of Y.
(iii). ?Cl(f-1 (IntT?(F))) ? f-1 (F) for every regular closed set F of Y.
(iv). ?Cl(f-1 (Cl(V)) ? f -1(ClT(V)) for every K-open set V of Y.
(v). f-1 (V) ? ?Int(f-1 (ClT?(V))) for every K-open set V of Y.
Proof. (i) ?(ii). Let B be any subset of Y. Assume that x?X~ f-1 (ClT (B)).Then f(x) ?Y~ ClT (B) and so there exists a K-open set V of Y containing f(x) such that V? B=?, so V? IntT? (ClT (B)))= ? and hence ClT? (V)? IntT? (ClT (B)))= ?. Therefore, there exists ? -open set U containing x such that f(U) ? ClT? (V).
Hence we have U? f-1(IntT? (ClT (B)))= ? and x?X~ ?Cl(f-1 (((IntT? (ClT (B)))) by Lemma 2.3.Thus we obtain ?Cl(f-1 (((IntT? (ClT (B)))) ? f-1 (ClT (B)).
(ii) ?(iii).Let F be any regular closed set of Y.Then F= ClT (IntT? (F) and we have ? Cl(f-1 ((IntT? (F)))= ? Cl(f-1 ((IntT? (ClT (IntT? t(F))))) ? f-1 (ClT (IntT? (F))) = f-1 (F).
(iii) ? (iv) . For any K-open set Vof X ClT (V) is regular closed .then ?- Cl(f-1(V)) ? ?Cl(f-1 (IntT?(ClT(V))?f-1(ClT (V))
(iv) ? (v) Let V be an K-open set of Y . the Y/ ClT?(V) is K-open set in Y and we have ?Cl(f-1(Y/ ClT?(V)) ?f-1(ClT (Y/ ClT?(V)) and hence X/ ?Int(f-1 (ClT?(V)) ? X/ f-1 (IntT (ClT? (V)) ?X/f-1(V).therefore we obtain f-1(V) ? ?Int(f-1 (ClT?(V)) .
4. Weakly*- quasi continuous
Now we define the regular in the topological space(X,T,T?) with some theorems Definition 4.1.A bitopological space ,(X,T,T?) is said to be regular if for each x?X and each T-open set U containing x, there exists a T-open set V such that x?V? ClT? (V) ?U.
Definition 4.2. A function f:(X,T,T?)? (Y,K,K?)) is said to be weakly*- quasi continuous(briefly.w*.q.c) if for every K -open set V of Y,
f -1 (ClT? (V)~V) is biclosed in X.
523Journal of Kerbala University , Vol. 10 No.3 Scientific . 2012