Πgβ-connectedness in Intuitionistic Fuzzy Topological Spaces

The paper aspires to discuss the basic properties of connected spaces. Also the concept of types of intuitionistic fuzzy πgβ-connected and disconnected in intuitionistic fuzzy topological spaces are introduced and studied. The research paper of topological properties is introducedby making the idea of being connected. It turns out to be easier to think about the property that is the negation of connectedness, namely the property of disconnectedness and separable. Also the concepts of intuitionistic fuzzy πgβC5-connectedness, intuitionistic fuzzy πgβCS-connectedness, intuitionistic fuzzy πgβCM-connectedness, intuitionistic fuzzy πgβ-strongly connectedness, intuitionistic fuzzyπ β-super connectedness and obtain several properties and some characterizations concerning connectedness in these spaces are explored.


Introduction
A predominant characteristic of a topological space is the concept of connectedness and disconnectedness. The former is one of the topological properties that is used to distinguish topological spaces. Connectedness [3] is a powerful tool in topology. Many researchers have investigated the basic properties of connectedness. The first attempt to give a precise definition of these spaces was made by Weierstrass who in fact instigated the notion of arc wise connectedness. However, the notion of connectedness which is used today was introduced by Cantor (1883) in general topology, Later onZadeh [12] introduced thenotionof fuzzysets. Fuzzytopological space was further developed by Chang [5]. Coker [6] introduced the intuitionistic fuzzy topological spaces. Connectedness in intuitionistic fuzzy special topological spaces was introduced byOscag and Coker [6]. Several types of fuzzy connectedness in intuitionistic fuzzy topological spaces were defined by Turnali and Coker [11] and studies these spaces very extensively and also delved into various generalization too of these spaces. Recently JenithaPremalatha and Jothimani [7] proposed herald into a new class of sets called πgβ-closed sets in intuitionistic fuzzy topological space, and these concepts have been used to define and analyse many topological properties. The aim of this paper is to study πgβ-connectedness and the notions of Intuitionistic fuzzy πgβ-separated sets, Intuitionistic fuzzy πgβ-connectedness and Intuitionistic fuzzy πgβdisconnectedness is dealt with in detail. Some of their types and their characterizations in Intuitionistic fuzzy topological spaces is studied. The problem focuses on the results when connectedness is replaced with πgβ-connectedness in intuitionistic fuzzy topological spaces.

. Then (i) A⊆B if and only if µ A (x) ≤ µ B (x) and ν A (x) ≥ ν B (x) for all x ∈X (ii) A = B if and only if A ⊆ B and B ⊆
3: [4] An intuitionistic fuzzy topology (IFT for short) on X is a family τ of IFSs in X satisfying the following axioms.
(i) 0~, 1~∈τ (ii) G 1 ∩ G 2 ∈τ for any G 1, G 2 ∈τ (iii) ∪G i ∈τ for any family {G i / i∈I}⊆τ. Definition 2.4: [4] Let (X,τ) be an IFTS and A =<x, µ A , ν A >be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by Int(A) = ∪{G / G is an IFOS in X and G ⊆ A} Cl(A) = ∩{K / K is an IFCS in X and A ⊆ K} Definition 2.5: An IF subset A is said to be IF regular open [8] The finite union of IF regular open sets is said to be IFπopen [8].
The complement of IFπ-open set is said to be IFπ-closed [8].
Definition 2.6: A is said to be IFβ-open [1] if A⊆Cl(Int(Cl(A))). The family of all IFβ-open sets of X is denoted by IFβO(X).
The complement of a IFβ-open set is said to be IFβ-closed [1]. The intersection ofall IFβ-closed sets containing A is called IFβ-closure [2] of A, and is denoted by IFβ-Cl(A).
The IFβ-Interior [2] ofA, denoted by IFβ-Int(A), is defined as union of all IFβ-open sets contained in A.
Definition 2.8( [6]): A mapping f: (X,τ)→(Y,σ) is called an intuitionistic fuzzy πgβ-irresolute if f -1 (V) is an IF-πgβ closed Setin (X,τ) for every IF-πgβ closed set V of (Y, σ). Definition 2.9( [10]): Two IFSs A andB in X are said to be q-coincident (AqB) if andonly if there exists an element x∈X Definition2. 10 ([10]): Two IFSs A and B in X are said to be not q-coincident (Aq C B) if and only if A⊆B C .

Intuitionisttic Fuzzy πGβ Connected Spaces and Its Types
Definition 3.1. [9]: Two subsets A and B in a IF space (X,τ) are said to be IF-πgβ -separated if and only if A∩πgβ-C l(B) =0 ∼and πgβ-C l(A)∩B= 0 ∼. If A andB are not IF πgβ-separated then it is said to be intuitionistic fuzzy πgβ-connected.
Remark: 3.1: Each two IFπgβ-seperated sets are always disjoint, since A∩B⊆A∩πgβ-Cl(B)= 0 ∼.  Proof: Let A and B are IFπgβ -separated sets. By Definition 3.1, A contains no IF πgβ -limit points of B. Then B contains all IFπgβ-limit points of B which are in A∪B and B is IFπgβ-closed in A∪B.
Therefore B is IFπgβ -closed in E. Similarly A is IF πgβclosed in E. Definition 3.3:A subset S of a space X is said to be IFπgβconnected relative to X if theredoes not exist two IF πgβseparated subsets A and B relative to X and S=A∪B, Otherwise S is said to be IFπgβ disconnected.
Definition 3.4: An intuitionistic fuzzy topological space (X,τ) issaid to be intuitionistic fuzzy πgβ-disconnected if there exists an intuitionistic fuzzy πgβ -open sets A, B in X, A≠0~, B≠0~, such that A∪B=1~ and A∩B=0~. If X is not IF πgβ-disconnected then it is said to be intuitionistic fuzzy πgβ-connected.
Theorem3. 4: Let A⊆B∪C such that A be a nonempty IFπgβ -connected set in a space X and B,C are IFπgβseparated. Then only one of the following conditions holds: (i) A⊆B and A∩C= 0 ∼. (ii) A ⊆ C and A∩B = 0 ∼. Proof: Since A∩C=0∼, then A⊆B. Also, if A∩B=0 ∼, then A⊆C. Since A⊆B∩C, then both A∩B=0∼ and A∩C=0 ∼cannot hold simultaneously. Similarly suppose thatA∩B= 0 ∼and A∩C=0 ∼then by Thorem 3.1 (i), A∩B and A∩C are IFπgβ-separated such that A=(A∩B)∪(A∩C) which contradicts with theI F π g β-connectedness of A. Hence one of the conditions (i) and (ii) must be hold.
Theorem 3.5: Let A and Bbe subsets in IF space (X,τ) such thatA ⊂ B ⊂IF πgβ-Cl(A). If A is IFπGβ-connected then B isIFπGβ -connected.
Proof: If B is IFπgβ -disconnected, then there exists two IFπgβ -separated subsets Uand V relative to X such that B =U∪V. Then either A⊆U or A⊆V. Let A⊆U. As A⊆U⊆B thenπgβ-C l B (A)⊆πgβ-C l B (U)⊆πgβ C l(U). Also πgβ-C l B (A) = B ∩πgβ C l(A) = B⊇πgβ C l(U). This implies B= πgβ-C l(U).
So U and V are not IFπgβ -separated and B is IFπgβconnected.
Therefore Eis IF πgβ-disconnected. Definition 3.5: An IFTS (X, τ) is said to be an intuitionistic fuzzy πgβ-connected space if the only IF sets which are bothintuitionistic fuzzy πgβ-open and intuitionistic fuzzy πgβ closed are 0∼ and 1∼. Definition 3.6: An IFTS (X, τ) in X is saidtobeIFπgβ C 5disconnected, if there exists an IF set A in X, which is both intuitionistic fuzzy πgβ open and intuitionistic fuzzy πgβ-closed, such that 0∼≠A≠1∼.
X is called C5 -connected, if X is not C5 -disconnected. Definition3.7( [10]):An IFTS (X, τ) is said to be an intuitionistic fuzzy C 5 -connected space if the only IFSs which are bothintuitionisticfuzzy open and intuitionistic fuzzy closed are 0∼ and 1∼.
Proof: Let (X,τ) be anintuitionistic fuzzy πgβ-connected space. Suppose (X,τ) is not an intuitionistic fuzzy GOconnected space, thenthereexists a properIFSA which both intuitionistic fuzzy open andintuitionistic fuzzy closed in (X, τ). That is A is both intuitionistic fuzzy πgβ-open andintuitionistic fuzzy πgβ-closedin (X,τ). This implies that (X,τ) is not an intuitionistic fuzzy πgβ-connected space. This is a contradiction. Therefore (X, τ) is an intuitionistic fuzzy GO-connected space. This implies B=A C ≠0∼, which is a contradiction to our hypothesis. Therefore, (X,τ) is an intuitionistic fuzzy πgβconnected space. Therefore there exist no non-zero intuitionistic fuzzy πgβopen sets A and B in (X,τ) such that A=B C ,B=(β-Cl(A)) C and A=(β-Cl(B)) C .
Sufficiency: Let A be both intuitionistic fuzzy πgβ-open and intuitionistic fuzzy πgβ-closed in (X,τ) such that 1∼≠A≠0∼. Now by taking B=A C , will lead to the contradiction to our hypothesis. Hence (X,τ) is an intuitionistic fuzzy πgβ-connected space. Definition 3.9: An IFTS (X,τ) is said to be an intuitionistic fuzzy πgβ-T 1/2 space if every intuitionistic fuzzy πgβ-closed set is an intuitionistic fuzzy closed set in (X,τ).

-connected space, then there exists a proper IF Set A which is both intuitionistic fuzzy πgβ open and intuitionistic fuzzy πgβ closed in (Y, σ). Since f is an intuitionistic fuzzy πgβ-continuous mapping, f -1 (A) is both intuitionistic fuzzy πgβ-open and intuitionistic fuzzy πgβ-closed in (X,τ). But this is a contradiction to hypothesis. Hence (Y, σ) is an intuitionistic fuzzy πgβ C 5 -connected space.
Theorem 3.13: If f: (X, τ)→(Y, σ) is an intuitionistic fuzzy πgβ-irresolute surjection and (X,τ) is an intuitionistic fuzzy πgβ-connected space, then (Y, σ) is also an intuitionistic fuzzy πgβ -connectedspace.
Proof: Suppose (Y, σ) is not an intuitionistic fuzzy πgβconnected space, then there exists a proper IFS A which is both intuitionistic fuzzy πgβ open and intuitionistic fuzzy πgβ-closed in (Y, σ). Since f is an intuitionistic fuzzy πgβirresolutemapping, f -1 (A) is bothintuitionistic fuzzy πgβ-open and intuitionistic fuzzy πgβ-closed in (X,τ). But this is a contradiction to hypothesis. Hence (Y, σ) is an intuitionistic fuzzy πgβ-connected space. Definition 3.10: An IFTS (X, τ) is called intuitionistic fuzzy πgβC 5 -connected between two IFSs A andB if thereis no intuitionistic fuzzy open set Ein (X,τ) such that A⊆E andEq C B.

Definition 3.11: An IFTS(X, τ) is called intuitionistic fuzzy πgβ connectedbetweentwo IFSs A andB if thereis no intuitionistic fuzzy πgβ open set E in (X, τ) such that A⊆E and Eq C B.
Theorem 3.14: If an IFTS (X, τ) is intuitionistic fuzzy πgβ connected between two IFSs A and B, then it is intuitionistic fuzzy πgβC 5 -connected between two IF Sets A and B.
Proof: Suppose (X, τ) is not intuitionistic fuzzy πgβ C 5connected between A and B, then there exists an intuitionistic fuzzy open set E in (X,τ) such that A⊆E and Eq C B. Since every intuitionistic fuzzy open set is intuitionistic fuzzy πgβopen set, there exists an intuitionistic fuzzy πgβ-open set E in (X, τ) such that A⊆E and Eq C B. This implies (X, τ) is not intuitionistic fuzzy πgβ-connectedbetween A and B, a contradiction to our hypothesis. Therefore (X, τ) is intuitionistic fuzzy πgβ C 5 -connected between A and B.

Theorem 3.15: An IFTS (X, τ) is intuitionistic fuzzy πgβconnected between two IFSs A and B if and only if there is no intuitionistic fuzzy πgβ-open and intuitionistic fuzzy πgβ-
closed set E in (X,τ) such that A⊆ E ⊆B C .
Proof: Necessity: Let (X,τ) be intuitionistic fuzzy πgβconnected between two IF Sets A and B. Suppose that there exists an intuitionistic fuzzy πgβopen andintuitionistic fuzzy πgβ-closed set Ein (X,τ) such that A⊆ E ⊆B C , then A⊆E and Eq C B. This implies (X, τ) is not intuitionistic fuzzy πgβ-connected between A and B, by Definition 3.11, A contradiction to our hypothesis. Therefore there is no intuitionistic fuzzy πgβ-open and intuitionistic fuzzy πgβclosed setE in (X,τ) such that A⊆ E ⊆B C .
Sufficiency: Suppose that (X, τ) is not intuitionistic fuzzy πgβ-connected between A and B. Then there exists an intuitionistic fuzzy-open set E in (X, τ) such that A⊆E andEq C B. Thisimplies that there is no intuitionistic fuzzyπgβopen set Ein (X, τ) such that A⊆ E ⊆B C .
But this is a contradiction to our hypothesis. Hence (X,τ) is intuitionistic fuzzy πgβ connected between A and B.
Proof:Suppose that (X, τ) is not intuitionistic fuzzy πgβ connected between A1 and B1, then by Definition 3.11, there exists an intuitionistic fuzzy πgβ openset E in (X, τ) such that A 1 ⊆E and Eq C B 1 . This implies E⊆B C and A1⊆E implies A⊆A1⊆E. That is A⊆E. Now let us prove that E ⊆B C , that is letus prove Eq C B. Suppose that EqB, then by Definition 2.9, there exists an element x in X such that µE (x) > νB (x) and νE (x) < µB (x). ThereforeµE (x)> νB (x) >νB 1 (x) and νE (x)< µB (x)< µB1 (x), since B⊆B 1 Thus EqB 1 . But E⊆B 1 . That is Eq C B 1 , which leads to a contradiction. Therefore Eq C B.
That is E⊆B C . Hence (X, τ) is not intuitionistic fuzzy πgβ connectedbetween A and B, which is a contradiction Thus (X, τ) is intuitionistic fuzzy πgβ connected between A1and B1.
Proof: Suppose (X,τ) is not intuitionistic fuzzy πgβ connectedbetween A and B. Then there exists an intuitionistic fuzzy πgβ open set Ein (X,τ) such that A⊆E and E⊆B C . This implies that A⊆B C , That is Aq C B. But this is a contradiction to our hypothesis.
Therefore (X,τ) is intuitionistic fuzzy πgβ connected between A and B.  Theorem 3.18: Let (X, τ) be an IFTS, then the following are equivalent.
(iv) There exists no intuitionistic fuzzy regular πgβ open sets A and B in (X,τ) such that A=0∼=B, A⊆B C .
(iii)⇒(iv) Let A and B be two intuionitic fuzzy regular πgβ open sets in (X,τ) such that A=0∼=B, A⊆B C.
If we take B=πgβ-cl(A)) C , then B is an intuitionistic fuzzy regular πgβ open set.

Applications
While focusing on some applications of connectedness, fixed point theorems in connection with application of connectedness. Fixed point theorems are useful in obtaining the (unique) solutions of differential and integral equations. In robotic motion planning, the connectedness of the configuration space conveys that one can reach the desired arrangement of solid objects from any initial arrangement.

Conclusion
The πgβ closed sets are used to introduce the concepts πgβ -connected space. Also, the characterization and thetypes of πgβ -connected spaces have been framed and analyzed. In general, the entire content will be a successful tool for the researchers for finding the path to obtain the results in the context of connected spaces in bi topologyand can be extended to Group theory. Also it is believed that this approach will prove useful for studying structures in the phase space of dynamical systems.