Contra qpi continuous functions in ideal bitopological spaces
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This academic article discusses contra qpI-continuous functions in ideal bitopological spaces. It begins with preliminaries on bitopological spaces, ideal topological spaces, and related concepts. It then defines contra qpI-continuous functions and establishes some of their properties, such as their relationship to contra qI-continuous functions and qpI-continuous functions. It proves several theorems about contra qpI-continuous functions and their images, including that the image of a qpI-connected space under a contra qpI-continuous function is connected. The article concludes by discussing contra qpI-irresolute mappings.
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
27
Contra QPI- Continuous Functions in Ideal Bitopological Spaces
Mandira Kar
Department of Mathematics, St. Aloysius College, Jabalpur (M.P.) 482001 India.
e-mail : karmandira@gmail.com
S. S. Thakur
Department of Applied Mathematics, Government Engineering College, Jabalpur (M.P.) 482011 India.
e-mail : samajh_singh@rediffmail.com
Abstract
In this paper, we apply the notion of qpI-open sets and qpI-continuous functions to present and study a new class
of functions called contra qpI-continuous functions in ideal bitopological spaces.
Keywords: Ideal bitopological space, qpI-open sets, qpI-continuous functions, qpI -irresolute functions.
AMS Subject classification: 54C08, 54A05
1. Preliminaries
In 1961 Kelly [6] introduced the concept of bitopological spaces as an extension of topological spaces. A
bitopological space (X, τ1, τ2) is a nonempty set X equipped with two topologies τ1 and τ2 [6]. The study of quasi
open sets in bitopological spaces was initiated by Dutta [1] in 1971. In a bitopological space (X, τ1, τ2) a set A of
X is said to be quasi open [1] if it is a union of a τ1-open set and a τ2-open set. Complement of a quasi open set is
termed quasi closed. Every τ1-open (resp. τ2-open) set is quasi open but the converse may not be true. Any union
of quasi open sets of X is quasi open in X. The intersection of all quasi closed sets which contains A is called
quasi closure of A. It is denoted by qcl(A) [9]. The union of quasi open subsets of A is called quasi interior of A.
It is denoted by qInt(A) [9].
Mashhour [10] introduced the concept of preopen sets in topology. A subset A of a topological space (X,
τ) is called preopen if A ⊂ Int(Cl(A)) . Every open set is preopen but the converse may not be true. In 1995,
Tapi [12] introduced the concept of quasi preopen sets in bitopological spaces. A set A in a bitopological space
(X, τ1, τ2) is called quasi preopen [12] if it is a union of a τ1-preopen set and a τ2-preopen set. Complement of a
quasi preopen set is called quasi pre closed. Every τ1-preopen (τ2-preopen, quasi open) set is quasi preopen but
the converse may not be true. Any union of quasi preopen sets of X is a quasi preopen set in X. The intersection
of all quasi pre closed sets which contains A is called quasi pre closure of A. It is denoted by qpcl(A). The union
of quasi preopen subsets of A is called quasi pre interior of A. It is denoted by qpInt(A).
The concept of ideal topological spaces was initiated Kuratowski [8] and Vaidyanathaswamy [13]. An
Ideal I on a topological space (X, τ) is a non empty collection of subsets of X which satisfies: i)A I and B ⊂
A ⇒ B I and ii) A I and B I ⇒ A∪B I If (X) is the set of all subsets of X, in a topological space
(X, τ) a set operator (.)*
: (X) → (X) called the local function [3] of A with respect to τ and I and is defined
as follows:
A∗
(τ, I) = {x∈XU ∩ A ∉ I, ∀ U∈ τ(x)}, where τ(x) =U∈ τ x∈U}.
Given an ideal bitopological space (X,τ1,τ2I) the quasi local function [3] of A with respect to τ1, τ2 and I denoted
by A∗
(τ1,τ2,I) ( in short A∗
) is defined as follows:
A∗
(τ1,τ2, I) = {x∈XU ∩ A ∉ I,∀ quasi open set U containing x}.
A subset A of an ideal bitopological space (X, τ1, τ2) is said to be qI- open [3] if A ⊂ qInt A∗
. A mapping f:
(X,τ1,τ2I) → (Y,σ1,σ2) is called qI-continuous [3] if f -1
(V) is qI-open in X for every quasi open set V of Y .
In 1996 Dontchev [2] introduced a new class of functions called contra-continuous functions. A function f:
X → Y to be contra continuous if the pre image of every open set of Y is closed in X.
Recently the authors of this paper [4 & 5] defined quasi pre local functions, qpI- open sets and qpI-
continuous mappings, qpI- irresolute mappings in ideal bitopological spaces.
Definition1.1. [4] Given an ideal bitopological space (X, τ1, τ2 I) the quasi pre local mapping of A with respect
to τ1, τ2 and I denoted by A∗
(τ1, τ2, I) (more generally as A∗
) is defined as follows: A∗
(τ1, τ2 ,I) = {x∈XU ∩
A ∉ I ,∀ quasi pre-open set U containing x}
Definition1.2. [4] A subset A of an ideal bitopological space (X, τ1, τ2, I ) is qpI- open if A ⊂ qpInt(A∗
).
Complement of a qpI- open set is qpI- closed. If the set A is qpI-open and qpI-closed, then it is called qpI-clopen
Definition1.3. [4] A mapping f: (X, τ1, τ2, I) → (Y, σ1 ,σ2) is called a qpI- continuous if f -1
(V) is a qpI- open set
in X for every quasi open set V of Y
Definition1.4. [5] A mapping f: (X,τ1,τ2,I) → (Y,σ1,σ2) is called qI- irresolute if f -1
(V) is a qI- open set in X for
every quasi open set V of Y.](https://image.slidesharecdn.com/contraqpi-continuousfunctionsinidealbitopologicalspaces-130830080630-phpapp02/85/Contra-qpi-continuous-functions-in-ideal-bitopological-spaces-1-320.jpg)
![Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
28
Definition1.5. [5] A mapping f: (X,τ1,τ2,I) → (Y,σ1,σ2) is called qpI- irresolute if f -1
(V) is a qpI- open set in X
for every quasi semi open set V of Y.
2. Contra qpI-continuous functions
Definition 2.1. A function f: (X, τ1, τ2, I) → (Y, σ1, σ2) is called contra qpI- continuous if f−1
(V) is qpI-closed in
X for each quasi open set V in Y.
Theorem 2.1. For a function f: (X, τ1, τ2, I) → (Y, σ1, σ2), the following are equivalent:
a) f is contra qpI-continuous .
b) For every quasi closed subset F of Y, f−1
(F) is qpI-open in X.
c) For each x X and each quasi closed subset F of Y with f(x) F, there exists a qpI-open subset U of X
with x U such that f (U) ⊂ F.
Proof: (a) ⇒ (b) and (b) ⇒ (c) are obvious.
(c) ⇒ (b) Let F be any quasi closed subset of Y. If x f−1
(F) then f(x) F, and there exists a qpI- open subset
Ux of X with x Ux such that f(Ux) ⊂ F. Therefore, f-1
(F) = ∪ {Ux: x ∈ f-1
(F)}. Hence we get f−1
(F) is
qpI-open. [4]
Remark 2.1. . Every contra qpI-continuous function is contra qI-continuous, but the converse need not be true
Example2.1. Let X ={ a, b, c } and I = {φ,{a}} be an ideal on X. Let τ1 = {X, φ,{c}}, τ2 = {X, φ, {a,
b}}, σ1 = {X, φ,{b}}and σ2 ={ φ, X} be topologies on X. Then the identity mapping f: (X,τ1, τ2 ,I) → (X,σ1,σ2) is
contra qI- continuous but not contra qpI- continuous as A= {b} is qI- open but not qpI-open in (X, τ1, τ2, I).
Theorem 2.2. If a function f: (X, τ1, τ2, I) →(Y, σ1, σ2) is contra qpI-continuous and Y is regular, then f is qpI-
continuous
Proof: Let x X and let V be a quasi open subset of Y with f(x) V Since Y is regular, there exists an quasi
open set W in Y such that f (x) W ⊂ cl(W) ⊂ V. Since f is contra qpI-continuous, by Theorem 2.1.there
exists a qpI-open set U in X with x U such that f (U) ⊆Cl (W). Then f (U) ⊆Cl (W) ⊆V. Hence f is qpI-
continuous [4].
Definition 2.2. A topological space (X, τ1, τ2, I) is said to be qpI -connected if X is not the union of two
disjoint non-empty qp I-open subsets of X.
Theorem 2.3. If f: (X, τ1, τ2, I) → (Y, σ1, σ2) is a contra qpI-continuous function from a qpI-connected space X
onto any space Y , then Y is not a discrete space.
Proof: Suppose that Y is discrete. Let A be a proper non-empty quasi clopen set in Y. Then f−1
(A) is a proper
non-empty qpI- clopen subset of X, which contradicts the fact that X is qpI-connected.
Theorem 2.4. A contra qpI-continuous image of a qpI-connected space is connected.
Proof: Let f: (X, τ1, τ2, I) → (Y, σ1, σ2) be a contra qpI- continuous function from a qpI-connected space X
onto a space Y . Assume that Y is disconnected. Then Y = A ∪ B, where A and B are non-empty quasi clopen
sets in Y with A ∩ B = ∅. Since f is contra qpI-continuous , we have that f−1
(A) and f−1
(B) are qpI-open non-
empty sets in X with f−1
(A) ∪ f−1
(B) = f−1
(A ∪ B) = f−1
(Y ) = X and f−1
(A) ∩ f−1
(B) = f−1
(A ∩ B) = f−1
(∅)
= ∅. This means that X is not semi-I-connected, which is a contradiction. Then Y is connected.
Definition 2.6. A mapping f: (X, τ1, τ2, I) → (Y, σ1, σ2) is called contra qpI- irresolute if f -1
(V) is a qpI- closed
set in X for every quasi semi open set V of Y.
Theorem 2.8. Let f: (X, τ1, τ2, I1) → (Y, σ1, σ2, I2) and g: (Y, σ1, σ2, I2) → (Z, ρ 1, ρ 2, I3) Then,
gof is contra qpI- continuous if g is continuous and f is contra qpI- continuous.
Proof: Obvious.
3. Conclusion
Ideal Bitopological Spaces is an extension for both Ideal Topological Spaces and Bitopological Spaces. It has
opened new areas of research in Topology and in the study of topological concepts via Fuzzy ideals in Ideal
Bitoplogical spaces. The application of the results obtained would be remarkable in other branches of Science
too.
References
1. Datta, M.C., Contributions to the theory of bitopological spaces, Ph.D. Thesis, B.I.T.S. Pilani, India.,
(1971)
2. Dontchev, J., Contra-continuous functions and strongly S-closed spaces, Internat. J. Math. Math. Sci.19,
(1996) 303–310
3. Jafari, S., and Rajesh, N., On qI open sets in ideal bitopological spaces, University of Bacau, Faculty of
Sciences, Scientific Studies and Research, Series Mathematics and Informatics., Vol. 20, No.2 (2010),
29-38](https://image.slidesharecdn.com/contraqpi-continuousfunctionsinidealbitopologicalspaces-130830080630-phpapp02/85/Contra-qpi-continuous-functions-in-ideal-bitopological-spaces-2-320.jpg)
Contra qpi continuous functions in ideal bitopological spaces
- 1. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 27 Contra QPI- Continuous Functions in Ideal Bitopological Spaces Mandira Kar Department of Mathematics, St. Aloysius College, Jabalpur (M.P.) 482001 India. e-mail : karmandira@gmail.com S. S. Thakur Department of Applied Mathematics, Government Engineering College, Jabalpur (M.P.) 482011 India. e-mail : samajh_singh@rediffmail.com Abstract In this paper, we apply the notion of qpI-open sets and qpI-continuous functions to present and study a new class of functions called contra qpI-continuous functions in ideal bitopological spaces. Keywords: Ideal bitopological space, qpI-open sets, qpI-continuous functions, qpI -irresolute functions. AMS Subject classification: 54C08, 54A05 1. Preliminaries In 1961 Kelly [6] introduced the concept of bitopological spaces as an extension of topological spaces. A bitopological space (X, τ1, τ2) is a nonempty set X equipped with two topologies τ1 and τ2 [6]. The study of quasi open sets in bitopological spaces was initiated by Dutta [1] in 1971. In a bitopological space (X, τ1, τ2) a set A of X is said to be quasi open [1] if it is a union of a τ1-open set and a τ2-open set. Complement of a quasi open set is termed quasi closed. Every τ1-open (resp. τ2-open) set is quasi open but the converse may not be true. Any union of quasi open sets of X is quasi open in X. The intersection of all quasi closed sets which contains A is called quasi closure of A. It is denoted by qcl(A) [9]. The union of quasi open subsets of A is called quasi interior of A. It is denoted by qInt(A) [9]. Mashhour [10] introduced the concept of preopen sets in topology. A subset A of a topological space (X, τ) is called preopen if A ⊂ Int(Cl(A)) . Every open set is preopen but the converse may not be true. In 1995, Tapi [12] introduced the concept of quasi preopen sets in bitopological spaces. A set A in a bitopological space (X, τ1, τ2) is called quasi preopen [12] if it is a union of a τ1-preopen set and a τ2-preopen set. Complement of a quasi preopen set is called quasi pre closed. Every τ1-preopen (τ2-preopen, quasi open) set is quasi preopen but the converse may not be true. Any union of quasi preopen sets of X is a quasi preopen set in X. The intersection of all quasi pre closed sets which contains A is called quasi pre closure of A. It is denoted by qpcl(A). The union of quasi preopen subsets of A is called quasi pre interior of A. It is denoted by qpInt(A). The concept of ideal topological spaces was initiated Kuratowski [8] and Vaidyanathaswamy [13]. An Ideal I on a topological space (X, τ) is a non empty collection of subsets of X which satisfies: i)A I and B ⊂ A ⇒ B I and ii) A I and B I ⇒ A∪B I If (X) is the set of all subsets of X, in a topological space (X, τ) a set operator (.)* : (X) → (X) called the local function [3] of A with respect to τ and I and is defined as follows: A∗ (τ, I) = {x∈XU ∩ A ∉ I, ∀ U∈ τ(x)}, where τ(x) =U∈ τ x∈U}. Given an ideal bitopological space (X,τ1,τ2I) the quasi local function [3] of A with respect to τ1, τ2 and I denoted by A∗ (τ1,τ2,I) ( in short A∗ ) is defined as follows: A∗ (τ1,τ2, I) = {x∈XU ∩ A ∉ I,∀ quasi open set U containing x}. A subset A of an ideal bitopological space (X, τ1, τ2) is said to be qI- open [3] if A ⊂ qInt A∗ . A mapping f: (X,τ1,τ2I) → (Y,σ1,σ2) is called qI-continuous [3] if f -1 (V) is qI-open in X for every quasi open set V of Y . In 1996 Dontchev [2] introduced a new class of functions called contra-continuous functions. A function f: X → Y to be contra continuous if the pre image of every open set of Y is closed in X. Recently the authors of this paper [4 & 5] defined quasi pre local functions, qpI- open sets and qpI- continuous mappings, qpI- irresolute mappings in ideal bitopological spaces. Definition1.1. [4] Given an ideal bitopological space (X, τ1, τ2 I) the quasi pre local mapping of A with respect to τ1, τ2 and I denoted by A∗ (τ1, τ2, I) (more generally as A∗ ) is defined as follows: A∗ (τ1, τ2 ,I) = {x∈XU ∩ A ∉ I ,∀ quasi pre-open set U containing x} Definition1.2. [4] A subset A of an ideal bitopological space (X, τ1, τ2, I ) is qpI- open if A ⊂ qpInt(A∗ ). Complement of a qpI- open set is qpI- closed. If the set A is qpI-open and qpI-closed, then it is called qpI-clopen Definition1.3. [4] A mapping f: (X, τ1, τ2, I) → (Y, σ1 ,σ2) is called a qpI- continuous if f -1 (V) is a qpI- open set in X for every quasi open set V of Y Definition1.4. [5] A mapping f: (X,τ1,τ2,I) → (Y,σ1,σ2) is called qI- irresolute if f -1 (V) is a qI- open set in X for every quasi open set V of Y.
- 2. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 28 Definition1.5. [5] A mapping f: (X,τ1,τ2,I) → (Y,σ1,σ2) is called qpI- irresolute if f -1 (V) is a qpI- open set in X for every quasi semi open set V of Y. 2. Contra qpI-continuous functions Definition 2.1. A function f: (X, τ1, τ2, I) → (Y, σ1, σ2) is called contra qpI- continuous if f−1 (V) is qpI-closed in X for each quasi open set V in Y. Theorem 2.1. For a function f: (X, τ1, τ2, I) → (Y, σ1, σ2), the following are equivalent: a) f is contra qpI-continuous . b) For every quasi closed subset F of Y, f−1 (F) is qpI-open in X. c) For each x X and each quasi closed subset F of Y with f(x) F, there exists a qpI-open subset U of X with x U such that f (U) ⊂ F. Proof: (a) ⇒ (b) and (b) ⇒ (c) are obvious. (c) ⇒ (b) Let F be any quasi closed subset of Y. If x f−1 (F) then f(x) F, and there exists a qpI- open subset Ux of X with x Ux such that f(Ux) ⊂ F. Therefore, f-1 (F) = ∪ {Ux: x ∈ f-1 (F)}. Hence we get f−1 (F) is qpI-open. [4] Remark 2.1. . Every contra qpI-continuous function is contra qI-continuous, but the converse need not be true Example2.1. Let X ={ a, b, c } and I = {φ,{a}} be an ideal on X. Let τ1 = {X, φ,{c}}, τ2 = {X, φ, {a, b}}, σ1 = {X, φ,{b}}and σ2 ={ φ, X} be topologies on X. Then the identity mapping f: (X,τ1, τ2 ,I) → (X,σ1,σ2) is contra qI- continuous but not contra qpI- continuous as A= {b} is qI- open but not qpI-open in (X, τ1, τ2, I). Theorem 2.2. If a function f: (X, τ1, τ2, I) →(Y, σ1, σ2) is contra qpI-continuous and Y is regular, then f is qpI- continuous Proof: Let x X and let V be a quasi open subset of Y with f(x) V Since Y is regular, there exists an quasi open set W in Y such that f (x) W ⊂ cl(W) ⊂ V. Since f is contra qpI-continuous, by Theorem 2.1.there exists a qpI-open set U in X with x U such that f (U) ⊆Cl (W). Then f (U) ⊆Cl (W) ⊆V. Hence f is qpI- continuous [4]. Definition 2.2. A topological space (X, τ1, τ2, I) is said to be qpI -connected if X is not the union of two disjoint non-empty qp I-open subsets of X. Theorem 2.3. If f: (X, τ1, τ2, I) → (Y, σ1, σ2) is a contra qpI-continuous function from a qpI-connected space X onto any space Y , then Y is not a discrete space. Proof: Suppose that Y is discrete. Let A be a proper non-empty quasi clopen set in Y. Then f−1 (A) is a proper non-empty qpI- clopen subset of X, which contradicts the fact that X is qpI-connected. Theorem 2.4. A contra qpI-continuous image of a qpI-connected space is connected. Proof: Let f: (X, τ1, τ2, I) → (Y, σ1, σ2) be a contra qpI- continuous function from a qpI-connected space X onto a space Y . Assume that Y is disconnected. Then Y = A ∪ B, where A and B are non-empty quasi clopen sets in Y with A ∩ B = ∅. Since f is contra qpI-continuous , we have that f−1 (A) and f−1 (B) are qpI-open non- empty sets in X with f−1 (A) ∪ f−1 (B) = f−1 (A ∪ B) = f−1 (Y ) = X and f−1 (A) ∩ f−1 (B) = f−1 (A ∩ B) = f−1 (∅) = ∅. This means that X is not semi-I-connected, which is a contradiction. Then Y is connected. Definition 2.6. A mapping f: (X, τ1, τ2, I) → (Y, σ1, σ2) is called contra qpI- irresolute if f -1 (V) is a qpI- closed set in X for every quasi semi open set V of Y. Theorem 2.8. Let f: (X, τ1, τ2, I1) → (Y, σ1, σ2, I2) and g: (Y, σ1, σ2, I2) → (Z, ρ 1, ρ 2, I3) Then, gof is contra qpI- continuous if g is continuous and f is contra qpI- continuous. Proof: Obvious. 3. Conclusion Ideal Bitopological Spaces is an extension for both Ideal Topological Spaces and Bitopological Spaces. It has opened new areas of research in Topology and in the study of topological concepts via Fuzzy ideals in Ideal Bitoplogical spaces. The application of the results obtained would be remarkable in other branches of Science too. References 1. Datta, M.C., Contributions to the theory of bitopological spaces, Ph.D. Thesis, B.I.T.S. Pilani, India., (1971) 2. Dontchev, J., Contra-continuous functions and strongly S-closed spaces, Internat. J. Math. Math. Sci.19, (1996) 303–310 3. Jafari, S., and Rajesh, N., On qI open sets in ideal bitopological spaces, University of Bacau, Faculty of Sciences, Scientific Studies and Research, Series Mathematics and Informatics., Vol. 20, No.2 (2010), 29-38
- 3. Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications 29 4. Kar, M., and Thakur, S.S., Quasi Pre Local Functions in Ideal Bitopological Spaces, Proceedings of the International Conference on Mathematical Modelling and Soft Computing 2012, Vol. 02 (2012),143- 150 5. Kar, M., and Thakur, S.S., On qpI-Irresolute Mappings, Internationa lOrganization of Scientific Research- Journal of Mathematics Vol. 03, Issue 2 (2012), 21-24 6. Kelly, J.C., Bitopological spaces, Proc. London Math. Soc.,13(1963), 71-89 7. Khan, M., and Noiri, T., Pre local functions in ideal topological spaces, Adv. Research Pure Math., 2(1) (2010), 36-42 8. Kuratowski, K., Topology, Vol. I, Academic press, New York., (1966) 9. Maheshwari, S. N., Chae G.I., and. Jain P.C., On quasi open sets, U. I. T. Report., 11 (1980), 291-292. 10. Mashhour, A. S., Monsef, M. E Abd El., and Deeb S. N, El., On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt., 53 (1982), 47-53 11. Tapi, U. D., Thakur, S. S., and Sonwalkar, A., On quasi precontinuous and quasi preirresolute mappings, Acta Ciencia Indica., 21(14) (2)(1995), 235-237 12. Tapi, U. D., Thakur, S. S., and Sonwalkar, A., Quasi preopen sets, Indian Acad. Math., Vol. 17 No.1, (1995), 8-12 13. Vaidyanathaswamy, R., The localization theory in set topology, Proc. Indian Acad. Sci., 20 (1945), 51-61
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