Download on almost contra x .continuous functions

Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 3, September 2011
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ON ALMOST CONTRA CONTINUOUS FUNCTIONS
M.TRINITA PRICILLA * and I.AROCKIARANI * *
*Department of Mathematics, Jansons Institute of Technology
Karumathampatti, India
[email protected]
**Department of Mathematics, Nirmala College for Women,
Coimbatore – 641 046.
ABSTRACT:
In this paper, we apply the notion of -closed sets in supra topological spaces to study
contra continuity and almost contra continuity as a new generalization of
a
continuity. Contra continuity is weaker than continuity,
continuity and contra continuity but stronger than almost
contra continuity.
INTRODUCTION
Dontchev [2] introduced notion of contra-continuity and obtained some results concerning compactness, S-closedness
and strong S-closedness in 1996.Ganster and Reilly [4] introduced a new class of functions called regular set-connected
functions in 1999.Jafari and Noiri [5 ] defined contra-pre continuous functions. Ekici [3] extended the notion to a class
of almost contra-pre continuous functions.
In 1970, Levine [6] introduced the concept of generalized closed sets in topological space and a class of
topological spaces called spaces. In 1983, A.S.Mashhour et al [7] introduced the notion of supra topological spaces
and studied S-S continuous functions and S* - continuous functions. In 2010, O.R.Sayed and Takashi Noiri [8]
introduced supra b - open sets and supra b - continuity on topological spaces. In this paper, we explore the notion of
closed sets to obtain the concept of contra continuous and almost contra continuous functions.
We also note that the class of contra continuous functions is properly placed between contra continuous
functions and almost contra continuous functions.
2. PRELIMINARIES
Definition: 2.1 [7]
A subclass ⊂ is called a supra topology on X if X and is closed under arbitrary union.(X, is
called a supra topological space (or supra space).The members of are called supra open sets.
Definition: 2.2 [7]
The supra closure of a set A is defined as Clµ !""#$%&'()"*+&,+ - .
The supra interior of a set A is defined as Int µ / !""#$%&)$*,&,+ 0 .
Definition 2.3 [8]
Let (1µ) be a supra topological space. A set A is called a supra b - open set if
A ⊆ Clµ (Int µ(A) ) ∪ Int µ(Cl µ(A)) .The complement of a supra b - open set is called a supra b - closed set.
Definition: 2.4 [1]
Let 1µ) be a supra topological space. A set A of X is called supra generalized b - closed set (simply gµ b closed) if bclµ(A) ⊆ U whenever A ⊆ U and U is supra open. The complement of supra generalized b - closed set is
supra generalized b - open set.
Definition: 2.5 [9]
A Subset A of (X, µ) is said to be supra regular open if A = Intµ(Clµ (A)) and supra regular closed if A =
µ
µ
cl (Int (A)) .
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M.TRINITA PRICILLA and I.AROCKIARANI
Definition: 2.6 [9]
Let 1µ) be a supra topological space. A set A of X is called supra generalized b - closed set (simply gµ b closed) if bc1µ(A) ⊆ U whenever A ⊆ U and U is µ -open. The complement of supra generalized b - closed set is
supra generalized b - open set.
Definition: 2.7 [9]
A function 2 1 3 41 5 is said to be gµ b –continuous if 2 67 8 is gµ b - closed in 1 for every
supra closed set V of 41 59
Definition: 3.1
3. Contra- -continuous functions
A function 2 3 4 is called contra -continuous if 2 67 8!" '()"*+ in 1 for each
supra open set 8)241 5.
Definition: 3.2
A function 2 1 3 41 5 is called
1. *%2*':(; '),:!,#)#" if 2 67 8 is '( )$*, in X for each supra open set V of Y.
2. <),:%& '),:!,#)#" if 2 67 8is supra closed in 1 for every supra open set V of 41 59
3. <),:%& '),:!,#)#" if 2 67 8is supra b-closed in 1 for every supra open set V of 41 59
4. <),:%& '),:!,#)#" if 2 67 8is closed in 1 for every supra open set V of 41 59
5. =*#(&% "*:'),*':*+ if 2 67 8 is '( )$*, in X for every 8 = >49
Theorem: 3.3
1. Every supra contra continuous is contra -continuous.
2. Every supra contra b- continuous is contra -continuous.
3. Every supra contra - continuous is contra -continuous.
Remark: 3.4
The converse of the above theorem is not true and it is shown by the following example.
Example: 3.5
Let X = {a, b, c}; {φ, , {a},{b},{a,b}}. Let 2 1 3 1 be the function defined by 2& '1 2 1 &,+2' &9Here f is contra -continuous.But 2 67 . . is not supra closed and
- closed. Therefore f is not ?@ABCD - continuous and contra - continuous.
Example: 3.6
Let X = {a, b, c}; {φ, , {a}} . Let 2 1 3 1 be an identity function. Here f is contra continuous.But 2 67 &. &. is not supra b- closed . Therefore f is not ?@ABCDE - continuous.
From the above theorem and definitions we have the following diagram :
'),:%& - continuous
contra - continuous
?@ABCDE - continuous
contra - continuous
Definition: 3.7
A space 1 is called -7FG space if every -closed set is E -closed .
Theorem: 3.8
If a function 2 1 3 41 5 is contra -continuous and if X is -7FG space, then 2 is ?@ABCDE continuous.
Proof: It is obvious.
Definition: 3.9
A space 1 is called -space if every -open set is supra open.
Theorem: 3.10
If a function 2 1 3 41 5 is contra -continuous and if X is -space, then 2 is ?@ABCD continuous.
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ON ALMOST CONTRA CONTINUOUS FUNCTIONS
Proof: It is obvious.
Definition: 3.11
A space 1 is called HIJ K -space if every -closed is -closed.
Theorem: 3.12
If a function 2 1 3 41 5 is contra -continuous and if X is HIJ K -space, then 2 is contra continuous.
Proof: It is obvious.
Theorem: 3.13
Suppose L ) is supra closed under arbitrary union. Then the following are equivalent for a function 2 3 4.
1. f is contra -continuous
2. For every supra closed subset F of Y,2 67 M L )9
3. For each N and each supra-closed set F in Y containing 2N, there exists a -open set U in X
containing N such that 2O P M.
Proof: 1Q R Let M*"#$%&'()"*+!,41 5 then 4 M!""#$%&)$*,!,41 5. By (1), 2 67 4 M 2 67 M!" '()"*+!,1 . This implies 2 67 M )$*,.
Therefore2 67 M L )1 .
2Q1 Let *"#$%&)$*,!,41 5 , then 4 8!""#$%&'()"*+!,41 5. By (2) 2 67 4 8 2 67 8!"L -open!,1 . This implies 2 67 8!"L -closed.Hence f is contra -continuous.
2Q3 Let F be supra-closed set in Y containing f(N). By (2), 2 67 M L )1 and N 2 67 M. Take O 2 67 M then 2O P M.
3Q2 Let M&,;"#$%&'()"*+)241 5 and N 2 67 M. From (3), there exists a -open set OS in X containing N
such that OS P 2 67 M. We have 2 67 M TSUVW OS . Thus 2 67 M is -open.
Lemma: 3.14
If Y is supra open in X, L ' Q L '4.
Proof: It is obvious.
Theorem: 3.15
2
If 2 1 3 41 5 is contra -continuous and U is supra open in X then XO Y O 3 4 is contra continuous .
Proof: Let V be supra closed in Y. Since 2 3 4 is contra -continuous,2 67 8 is -open in X.
67
67
2
2
Z XO[ 8 2 67 8 O is -open in X. By lemma 3.14,Z XO[ 8 is -open in U.
Lemma: 3.16
The following properties are equivalent for a subset A of a space X.
1. A is '( )$*, .
2. A is -closed and %*#(&% )$*,9
3. A is -closed and )$*,9
Proof: It is obvious.
Theorem: 3.17
For a function 2 3 4 the following properties are equivalent.
1. f is $*%2*':(; -continuous.
2. f is contra -continuous and %*#(&% )$*, continuous.
3. f is contra -continuous and -continuous.
Proof: It is obvious.
4.
Definition: 4.1
8 = )41 5.
Almost contra- -continuous functions
A function 2 3 4 is said to be almost contra -continuous if 2 67 8 '1 for each
Definition: 4.2
A function 2 1 3 41 5 is called
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M.TRINITA PRICILLA and I.AROCKIARANI
1.
2.
(\)":gµ b –continuous function if 2 67 8is open in 1 for every supra regular open set V
of 41 59
] open function if image of each open set is open.
Theorem: 4.3
Let 1 and 41 5 be supra topological spaces.The following statements are equivalent for a function
2 34
1. f is almost contra- -continuous function.
2.2 67 M )1 for every M = '41 5.
3. For each N and each %*#(&% -closed set F in Y containing 2N, there exist a -open set U
in X containing N such that 2O P M.
4. For each N and each %*#(&% -open set V in Y not containing 2N, there exist a -closed
set K in X not containing N such that 2 67 8 P ^.
5. 2 67 !,: _'( L` ' for every supra open subset G of Y.
6.2 67 '( _a,: M` L ) for every supra closed subset F of Y.
Proof: 1Q2. Let M = '41 5 then 4 M = )41 5. By (1), 2 67 4 M 2 67 M L '1 . This
implies 2 67 M L )1 .
2Q1 Let 8 = )41 5 then 4 8 = '41 5. By (2) 2 67 4 8 L -open. This implies
X-2 67 8 L -open. Hence 2 67 8 L -closed.
2Q3 Let F be any %*#(&% -closed set in Y containing f(N). By (2), 2 67 M L )1 and N 2 67 M. Take
O 2 67 M then 2O P M.
3Q2 Let M = '41 5 and N 2 67 M. From (3), there exists a -open set OS in X containing N such that
OS P 2 67 M. We have 2 67 M TSUVW OS . Thus 2 67 M is -open.
3Q4 Let V be a %*#(&% open set in Y not containing f(N). Then Y-V is a %*#(&% -closed set containing f(N). By
(3), there exist a -open set U in X containing N such that 2O P 4 8. Hence O P 24 8 P 2 67 8 and
then 2 67 8 P O. Take ^ O. We obtain a -closed set K in X not containing N.
4Q3 Let F be a %*#(&% open set in Y containing f(N). Then YF is a %*#(&% -open set in Y not containing f(N).
By (4), there exist a -closed set K in X not containing N such that 2 67 4 M P ^9 This implies X2 67 M P
^ Q ^ P 2 67 M Q 2 ^ P M9 Take O ^ then U is -open set in X containing N such that
2O P M9
1Q5 Let G be supra open subset of Y. Since !,: _'( L` is %*#(&% -open, then by (1)
2 67 !,: _'( L` L '1 .
5Q1 Let 8 = )41 5, then V is supra open in Y. By (5) 2 67 !,: _'( 8` L '1 .
2Q6 It is obtained similar as (1)b (5) .
Example: 4.4
Let X={a,b,c};={c1 1 &.. and 5={c1 1 &.1 .1 &1 ... Here the function 2 1 3 1 5 is defined as f(a)=c;
f(b)=b and f(c)=a. Here f is almost contra -continuous function but not %*#(&% set connected.
Example: 4.5
Let X={a,b}, ={c1 1 &.. and 5={c1 1 ... Then the identity function 2 1 3 1 5 is
'),:%& '),:!,#)#" and %*#(&% set connected but not $*%2*':(; '),:!,#)#".
Remark: 4.6
From the above theorem and examples we have the following diagram
1
2
3
4
5
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ON ALMOST CONTRA CONTINUOUS FUNCTIONS
Here the numbers 1-5 represent the following :
1.*%2*':(; '),:!,#)#" function 2. '),:%& '),:!,#)#" function 3. '),:%& '),:!,#!:;
5. Almost contra- -continuity
4. =*#(&% set connected
Theorem: 4.7
Suppose that -closed sets are supra closed under finite intersection. If 2 3 4 is almost contra- 2
continuous function and A is -open subset of X, then the restriction X Y 3 4 is almost contra- continuous.
Proof: Let M = '4. Since f is almost contra -continuous then 2 67 M L )1 . Since A is -open
67
2
2
in X it follows that Z X[ M 2 67 M L )1 . Therefore, X is almost contra- -continuous
function.
Theorem: 4.8
Let 2 3 4 and 4 3 d be a function. Then, the following properties hold:
1.
If f is almost contra - -continuous and g is %*#(&% set-connected, then e 2 3 d is almost contra -continuous and almost- -continuous functions.
2.
If f is almost contra - -continuous and g is fgChg?Bij continuous, then e 2 3 d is -continuous
and contra- -continuous functions.
3.
If f is almost contra - -continuous and g is %*#(&% set-connected, then e 2 3 d is almost contra -continuous and almost- -continuous functions.
Proof: It is obvious.
Theorem: 4.9
Let 2 3 4 and 4 3 d be a function. Then the following properties hold:
1. If f is almost -continuous and g is fgChg?Bij continuous function, then e 2 3 d is contra- continuous.
2. If f is almost - -continuous and g is %*#(&% set-connected, then e 2 3 d is almost contra- continuous.
Proof: It is obvious.
Theorem: 4.10
If 2 3 4 is a surjective M- -open function and 4 3 d is a function such that e 2 3 d is
almost contra-- -continuous, then g is almost contra-- -continous.
Proof: Let V be any %*#(&% closed set in Z. Since e 2 is almost contra- -continuous, e 267 8 L )1 . Since f is surjective, M- -open map, 2 e 267 8 2 Z2 67 _67 8`[ 67 8 is open.Therefore g is almost contra- -continuous.
Theorem: 4.11
If 2 3 4 is a surjective M- -closed and 4 3 d is a function such that
e 2 3 d is almost contra- -continuous, then g is almost contra- -continuous
Proof: It is obvious.
Theorem: 4.12
If a function 2 3 4 is almost contra- -continuous and &(\)": continuous then 2 is %*#(&% set connected.
Proof: Let 8 = )4. Since 2 is almost contra- -continuous and almost continuous 2 67 8 is -closed and
supra open. Hence 2 67 8 is '( )$*, . Hence f is %*#(&% set connected.
REFERENCES:
[1] I. Arockiarani and M.Trinita Pricilla, “On Supra generalized b-closed sets”, Antarctica
Journal of Mathematics, Volume 8(2011).
[2] J.Dontchev, “Contra-continuous functions and strongly-S-closed spaces”,
Internat.J.Math.Math.Sci.19 (2)(1996),303-310.
[3] E.Ekici, “Almost contra-pre continuous functions”, Bull.Malays.Math.Sci.Soc, 27: (2004) 53-65.
[4] M.Ganster and I.Reilly, “More on almost-s-continuity”, Indian J.Math , 41: (1999) 139-146.
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M.TRINITA PRICILLA and I.AROCKIARANI
[5] S.Jafari and T.Noiri, “Contra-k-continuous functions between topological spaces”, Iran.Int.J.Sci.
2(2) (2001), 153-167.
[6] N.Levine,” Generalized closed sets in topology”, Rend.Circ. Mat.Palermo (2)19(1970),89-96.
[7] A.S. Mashhour, A.A. Allam, F.S. Mahamoud and F.H.Khedr, “On supra
topological spaces”, Indian J.Pure and Appl. Math. No. 4, 14 (1983), 502 – 510.
[8] O.R. Sayed and Takashi Noiri, “on supra b - open sets and supra b –Continuity on topological
Spaces”, European Journal of pure and applied Mathematics, 3(2) (2010), 295 – 302.
[9] M.Trinita Pricilla and I. Arockiarani ,”on supra T-closed sets”, International journal of
Mathematical Archive,2-256(Accepted).
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