Čech Closure space was introduced by E. Čech [ 3] in 1966. Fuzzy Čech Closure space was first introduced by the A.S. Mashhour and M.H. Ghanim [ 6]. The notions of closure system and closure operator are very useful tools in several areas of classical mathematics. They play an important role in topological spaces, Boolean algebra, convex sets, etc.

Biclosure space was introduced by Chandrasekhar Rao, Gowri, and Swaminathan [ 8]. Fuzzy biclosure space was introduced by Tapi U.D. and Navalakhe R. [ 11]. Such spaces are equipped with two arbitrary fuzzy closure operators. He extended some of the standard results of separation axioms in fuzzy closure space to fuzzy biclosure space. Thereafter a large number of papers have been written to generalize the concept of fuzzy closure space to fuzzy biclosure space. In this paper, the authors have introduce fuzzy connectedness in fuzzy biclosure space and studied some of its fundamental properties.

1.1 Definition [ 1]

An operator u : P(x) →P(x) defined on the power set P(X) of a set X satisfying the axioms:

• u Ф =Ф;
• A ⊆uA, for every A ⊆X;
• u(A ∪B) = u A ∪uB, for all A,B ⊆X.

is called a Čech closure operator and the pair (X, u) is a Čech closure space.

1.2 Definition [ 2]

Two maps u₁ and u₂ from power set of X to itself are called biclosure operators for X if they satisfy the following properties:

• u₁Ф=Ф,u₂Ф =Ф
• A ⊆u₁ A, A ⊆u₂ A, for all A ⊆X;
• u₁ (A ∪B) = u₁A ∪u₁B, u₂ (A∪B) = u₂A∪u₂B, for all A⊆X.

A structure (X, u₁, u₂) is called a biclosure space

1.3 Definition [ 9]

A closure space (X, u) is connected if there exists any continuous mapping f from X to the discrete space {0, 1} is constant. A subset A in a closure space (X, u) is said to be connected if A with the subspace topology is a connected space.

1.4 Definition [ 5]

Let (X, u₁, u₂) and (Y, v₁, v₂) be biclosure spaces and let i ∈{1,2}. Then a mapping f: (X, u₁, u₂) →(Y, v₁, v₂) is called

(i) i -open (respectively, i -closed) if the mapping f: (X, u_i) →(Y, v_i) is open (respectively, closed).

(ii) Open (respectively, closed) if f is i-open (respectively, i-closed) for all i ∈{1,2}.

(iii) i -continuous if the mapping f: (X, u_i) →(Y, v_i) is continuous.

(iv) Continuous if f is i-continuous for all i∈ {1,2}.

1.5 Definition [ 4]

Let X is a non empty fuzzy set. A function k: I^x→ I^x is called fuzzy Čech closure operator on X if it satisfies the following conditions

• k(0) =0;
• λ≤k(λ, for all λ∈ I^x
• k (λ₁νλ₂) = k (λ₁)νk (λ₂) for all λ₁,λ₂∈ I^x

The pair (X, k) is called fuzzy Čech closure space.

1.6 Definition [ 7]

Let (X, u) and (Y, v) be fuzzy closure spaces. A mapping f: (X, u)→ (Y, v) is said to be fuzzy continuous if f(u(χ_{λ}))≤v{f(χ_{λ})}for every fuzzy subset λ≤I^x

1.7 Definition [ 10]

Let X is a nonempty fuzzy set. A mapping k: I^x→ I^x is called fuzzy Čech closure operator on X. A fuzzy Čech closure space (X, k) is said to be fuzzy connected if and only if any F-continuous mapping f from X to the fuzzy discrete space {0, 1} is constant.

1.8 Definition [ 6]

• u₁{0} = 0 and u₂ {0} = 0;
• λ≤u₁λand λ≤u₂λfor all λ≤I^x.
• u₁(λ₁νλ₂) = u₁{λ₁} νu₁{λ₂} and u₂(λ₁νλ₂) = u₂{λ₁} νu₂{λ₂} for all λ₁,λ₂≤I^x.

1.9 Definition [ 6]

Let (x, u₁, u₂) and (Y, v₁, v₂) be fuzzy biclosure spaces and let i∈ {1, 2}. A map f: (X, u₁, u₂) (Y, v₁, v₂) is called fuzzy i- continuous if the mapping f: (X, u_i) (Y, v_i) is fuzzy continuous. A mapping f is called fuzzy continuous if f is fuzzy i-  continuous for each i∈{1, 2}.

2.1 Definition

Let X is a nonempty fuzzy set and k_i : I^x→ I^x for i = {1, 2} is called fuzzy closure operators on X. A fuzzy biclosure space i (x, k₁, k₂) is called fuzzy connected if and only if any F-continuous mapping f from X to the fuzzy discrete space {0, 1} is constant.

2.2 Example

Let X= {a, b, c} is a non empty set. Define fuzzy closure operator k₁ : I^x→I^x such that,

Hence (X, k₁ ) is a fuzzy closure space

FOS (X, k₂)

For another fuzzy closure operator k₂ : I^x→ I^x such that,

Hence (X, k₂) is a fuzzy closure space.

FOS (X, k₂)

So here (X, k₁ , k₂ ) is a fuzzy biclosure space.

Fuzzy open sets of fuzzy biclosure space (X, k₁, k₂ )

We define an F-continuous mapping f:X→{0,1} such that

Here mapping f is constant. Hence (X, k₁ , k₂ ) is a fuzzy connected fuzzy biclosure space.

2.3 Example

Let X = {a, b, c, d} is a non empty set and consider a fuzzy closure operator which is defined by k₂ : I^x→I^x such that,

For all fuzzy subsets λ≤I^x let

Hence (X, k₁) is a fuzzy closure space.

Fuzzy open sets of(X,k₁)

Consider another fuzzy closure operator k₂ : I^x→I^x such that 

For all fuzzy subsets χ_λ≤I^x let,

Hence (X, k₂ ) is a fuzzy closure space.

Fuzzy open sets of fuzzy closure space,

(X,k₂)

Here (x, k₁ , k₂ ) is a fuzzy biclosure space.

Fuzzy open sets of fuzzy biclosure space (x, k₁ , k₂ ) are,

Let f:X→{0,1} is a F-continuous mapping such that,

Here F-continuous mapping f is constant.

Hence (X, k₁ , k₂ ) is a fuzzy connected fuzzy biclosure space.

3.1 Proposition

A fuzzy biclosure space (X, k₁ , k₂ ) is fuzzy connected if and only if it has no separation. 

Proof

If (X, k₁ , k₂ ) is fuzzy connected fuzzy biclosure space and it has a separation . Define f:X→{0,1} by ifχ_{x}≤χ_{y} . Then F-mapping f is continuous but not constant. It contradicts the fuzzy connectedness of fuzzy biclosure space (X, k₁ , k₂ ).

Hence fuzzy biclosure space (X, k₁ , k₂) has no separation.

Conversely, consider fuzzy biclosure space (X, k₁ , k₂ ) has no separation, if X is not fuzzy connected. Then there exists an F-continuous map f:X→{0,1} is not constant.f^-1(0) and f^-1(1) is non-empty and ,

Thus fuzzy biclosure space X has a separation f-¹(0),f^-1(1). Hence fuzzy biclosure space (X, k₁ , k₂ ) is fuzzy connected.

3.2 Proposition

An F-continuous image of a fuzzy connected fuzzy biclosure space is fuzzy connected.

Proof

Let f : X→ Y be an F-continuous surjective map where X, Y are fuzzy biclosure spaces and X is fuzzy connected fuzzy biclosure space. Suppose fuzzy biclosure space Y is not fuzzy connected, and then by definition, F-continuous function f is not constant. Y has a separation, say χ_u|χ_v . Let , then are disjoint fuzzy open sets of X. This contradicts the fact that X is fuzzy connected fuzzy biclosure space. Therefore Y must be fuzzy connected fuzzy biclosure space.

3.3 Proposition

The union of any family of fuzzy connected sets of fuzzy biclosure space (X, k₁ , k₂ ) with a common point is fuzzy connected.

Proof

Let χ_{xa} be a family of fuzzy connected sets and for all α. Let be any fuzzy continuous map and be the restriction of f to . Since f is F-continuous, each f_α is F-continuous and X_α is fuzzy connected so f_α is constant. Now let for all α and for all α and , i.e. f is constant. Thereforeχ_{UXα} is fuzzy connected set of fuzzy biclosure space (X, k₁ , k₂ ).

3.4 Proposition

Let χ_{λ1}and χ_{λ2}are fuzzy subsets of a fuzzy connected fuzzy biclosure space (X, k₁ , k₂ ) such that, and   is fuzzy connected, and then χ_{λ2}is fuzzy connected.

For each point χ_{p} in a fuzzy connected fuzzy biclosure space (X, k₁ , k₂ ) the component. of X is the largest fuzzy 1 2 connected set in X which contains the point χ_{p}.

3.5 Proposition

For each point χ_{p} in a fuzzy connected biclosure space (X, k₁ , k₂ ), the component χ_{C(p)} is a fuzzy closed set.

In this paper, the idea of fuzzy connectedness in fuzzy biclosure space was introduced. Fuzzy connectedness is very useful in image segmentation. In image segmentation, fuzzy connectedness is used in fuzzy topological space. Fuzzy closure space is the generalization of fuzzy topological space. Fuzzy connectedness in fuzzy closure space gives strength of connectedness between every two pairs of image pixels is considered. Fuzzy connectedness is also useful in digital image processing which is helpful in medical sciences.