On Contra (δ, gδ)–Continuous Functions
Zainab Aodia Athbanaih
University of Al-Qadissiya, College of Education
Abstract
In 1980 T . Noiri introduce δ – sets to define a new class of functions called δ–continuous and in 1996 Dontchev and Ganster introduce and studied a new class of sets called δ–generalized closed and in 2000 Dontchev and others introduce – closed sets. In this paper we use these sets to define and study a new type of functions called contra –continuous as a new type of contra continuous the functions contra δ–continuous is stronger than contra continuous function , contra – continuous functions and weaker than Rc–continuous and each one of them are independent with - continuous functions.
Preservation theorems of this functions are investigated also several properties concerning this functions are obtain . The relationships between the δ –closed set with the other sets that have so important to get many results are investigated . finally we study the relationships between this functions with the other functions .
الخلاصة
فـي عـام 1980 قدم تكاشي نويرا المجمـوعـات – δ ليعـرف نوع جديد من الدوال هي الدوال المستمرة – δ . وفي عام 1996 Dontchev و Ganster قدموا ودرسوا نوع جديد من المجموعات المغلقة هي ) δ – المعممة المغلقة ( وفي عام 2000 أيضاً Dontchev وباحثين اخرين قدموا المجموعات المغلقة – . في هذا البحث استخدمنا هذه المجموعات لتعريف نوع جديد من الدوال وهي الدوال الضد مستمرة – كنوع جديد من الدوال الضد مستمرة ( او العكس مستمرة ). حيث ان الدالة الضد مستمرة – δ هـي اقـوى من الدوال الضد مستمرة والضد مستمرة - gδ واضعف من الدوال المستمرة – Rc] وكل واحدة منهما مستقلة عن الدوال المستمرة – δ ) و ( . وللحفاظ على مبرهنات عن تلك الدوال تم الاستقصاء او التحري عنها كذلك حصلنا على بعض الخواص المتعلقة بتلك الدوال ، علاقة المجموعات المغلقة – δ بمجموعات اخرى لها اهمية في الحصول على بعض النتائج . وأخيراً درسنا علاقة هذه الدوال مع دوال اخرى .
1. Introduction
In 1996, Dontchev introduce a new class of functions called contra – continuous functions, in 1999 Jafari and Noiri introduce and studied a new functions called contra super – continuous, and in 2001 they present and study a new functions called contra - continuous .
In this paper we introduce the notion of contra - continuous, where contra δ–continuous functions is stronger than both contra–continuous and contra - continuous. We astablish several properties of such functions . especially basic properties and preservation theorems of these functions are investigated . moreover , we investigate the relationships between δ–closed sets and other sets also the relation ships between these functions with the other functions .
2. Preliminaries
Through the present note and (or simply and ) always topological spaces .
A subset of a space is said to be regular open (resp. regular closed , δ – open, - closed, clopen, - closed) if (resp. , where is [Noiri,1980] regular open , if whenever and is open [Dontchev,2000] , open and closed , if whenever and is δ – open [Dontchev,2000] . Apoint is said to be δ – cluster point of aset if , for every regular open set containing . The set of all δ – cluster points of is called δ – closure of and denoted by . if , then is called δ – closed [Noiri,1980 ], or equivalently is called δ– closed if , where is regular closed
The collection of all δ – closed ( resp . δ – open , - closed , - clopen , - closed , regular closed ) sets will denoted by ( resp . , , , , ) also the complement of δ – open ( resp . - closed , - closed , regular closed ) sets is called δ – closed ( resp . - open , - open , regular open ) .
Its worth to be noticed that the family of all δ – open subsets of a space is a topology on which is denoted by - space and some time is called semi – regularization of . As a consequence of definitions we have .
finally anon – empty topological space is called hyper connected or irreducible [Dontchev,2000 ] if every non – empty open subset of is dense .
3. Properties of - Closed Sets
proposition 3.1
Let be a subset of aspace :
1- If is regular open , then is open .
2- If is δ – closed , then is closed .
Proof :-
1- Is clearly
2- Let be a δ – closed set , is δ – open
, since is regular so by part ( 1 ) , is open and countable unions of open sets are open , is open , so , is closed .
Remark 3.2
The converse of Proposition (3.1) is not true in general . To see this we give the following example .
Example 3.3
Let and be a topology defined on .
Let , is closed set but not δ – closed .
Since not δ – open .
Also , is open but not regular open .
Proposition 3.4
Let be a subset of a space .
1- If is regular open , then is δ – open .
2- If is δ – closed , then is - closed .
3- If is closed , then is - closed .
4- If is - closed , then is - closed .
Proof :-
1- Let be a regular open set , is δ – open
2- Let , is δ – closed , so
, so , therefore for every , is regular open and , , this means that , for every regular open in and since is δ – closed so by lemma ( 3.1 ) part ( 2 ) .
is closed so . and is δ – open by part ( 1 ) thus , then , for every δ – openset .
Therefore is - closed .
3- Let , is closed , let , then , , , , this means that , and is open , also since is closed , , thus , when and is open .
Therefore is - closed .
4- Let be a - closed set , so , where is open and , by part (2) from lemma ( 3.1 ) there exist δ – open set , such that and , so , Thus , when , is δ – open in .
Therefort is - closed .
Remark 3.5
The converse of lemma ( 3.4 ) is not true in general , to see this,we give the following counter example .
Examples 3.6
1- Let and be a topology defined on .
Let is - closed but not δ – closed . Also is - closed but not closed .
2- Let and be a topology defined on .
Let is - closed but not - closed .
Lemma 3.7 [Dontchev,2000 ]
Let be a topological space , and be alocally finite family , :
f is - closed sets , , then is also - closed .
Proposition 3.8
Let be a topological space , and be alocally finite family , :
If is - closed set , , then is also - closed .
Proof :-
Let is δ – closed set , , is δ – closed this means that , , so , since by lemma 3.1 part ( 2 ) is closed so . Thus so , therefort is δ – closed .
Proposition 3.9
The following properties are equivalent for a subset of a space :
1- is clopen .
2- is regular open and regular closed .
3- is δ – open and δ – closed .
4- is δ – open and - closed .
Proof :-
1 → 2 : clearly since is open and closed .
2 → 3 : by proposition ( 3.4 ) part ( 1 ) .
3 → 4 : by proposition( 3.4 ) part ( 2 ) .
4 → 3 : is δ – open from part ( 4 ) , let is - closed set so , when and is δ – open by proposition ( 3.4 ) ( 1 ) , there exist , is regular open such that , since when , is regular open in .
Thus , when , is regular open , therefore is δ – closed .
3 → 2 and 2 → 1 : clearly by proposition ( 3.1 )
Lemma 3.10 [Dontchev,2000]
For a topological space the following conditions are equivalent :
1- is - space .
2- is almost weakly hausdorff and semi – regular [ NavalaG,2000 ] .
Lemma 3.11
Let be a topological space , :
1- If is closed , then is δ – closed .
2- If is - closed , then A is δ – closed .
Proof :-
1- Let be a closed set , - space , since so , open , where , so , therefort is δ – closed .
2- clearly
Proposition 3.12
Let be an open or dense ( resp . δ – closed ) subset of a space , and ( resp . ) then ( resp . ) .
Proof :-
Clearly that if is dense set .
To prove that , let , so , , if is open in ,
So
Also so
Therefore
To prove that , let and , let , so
, since , , therefore
and .
, this means that for every , and so and also , this means that for every open in and , , so and , by proposition ( 3.1 ) ( 1 ) , so and , so . Thus
by proposition ( 3.4 ) ( 1 ) , therefore , when ever . Thus .
proposition 3.13
Let , if ( resp. ) and ( resp . ), then ( resp . ) .
Proof :-
To prove that
Let , , let , where . Then . Since is - closed relative to .
We have , so .
Since , it follows that , since then .
But . Therefore which implies that , . Thus .
To prove that suppose that so and
, thereforte
Lemma 3.14 [Dontchev,2000]
If is almost weakly hausdorff space , let is - closed then it is closed set .
Lemma 3.15 [Dontchev,2000]
For anon – empty topological space the following conditions are equivalent :
1- is hyperconnected .
2- Every subset of is - closed and is connected .
4. Continuity of contra - continuous functions
Definition 4.1
A function is called :
1- contra δ – cont . ( resp . contra - cont . ) at a point , if for each open subset in containing , there exists a δ – closed ( resp . - closed ) subset in containing such that .
2- contra δ – cont . ( resp . contra - cont . ) if it have this property at each point of .
Example 4.2
Let , and are topological spaces defined on and respectively , let be the Identity function .
Note that is contra - continuous functions .
Proposition 4.3
For a function , the following are equivalent :
1- is contra δ – cont . ( resp . contra - cont . ) .
2- for every open set of , ( resp . ).
3- for every closed of , ( resp . ) .
Proof :-
1 → 2 : Let be an open subset of , and let , since , there exists ( resp . containing such that . We obtain that by lemma ( 3.7) is δ – closed ( resp . , - closed ) in .
2 → 3 : Let be a closed subset of . Then is open , by part ( 2 ) , is δ – closed ( resp . - closed ) , thus is δ – open ( resp . - open ) in
3 → 1 : Let be a closed subset of containing , from part ( 3 ) , is δ – open ( resp . - open ) , take . Then . Therefore is contra δ – cont . ( resp . contra - cont . ) .
Proposition 4.4
Let be contra δ – cont . ( resp . contra - cont . ) functions , if or dense ( resp . ) , then is contra δ – cont . ( contra - cont . )
Proof :-
To prove that is contra – cont .
Let be aclosed in . , since is contra – cont . , . By proposition (3.12) , thus is contra – cont .
Remark 4.5
The converse of proposition ( 4.4 ) is not true in general . To see this, we give the following counter example .
Example 4.6
Let , and are topological space defined on and respectively , let be the Identity function .
Let , see that is contra - cont . But is not contra - cont . function .