The concept of fuzzy sets was introduced by Zadeh [1]. Chang [2] defined fuzzy topological spaces and several authors have since studied and generalized this concept. One such generalization was introduced by Sostak [3], who used the concept of degree of openness. This type of generalization was rephrased by Chattopadhyay et al. [4] and Ramadan [5].

As a further generalization of fuzzy sets, Atanassov [6] introduced the concept of intuitionistic fuzzy sets and Coker [7] defined the intuitionistic fuzzy topological spaces using these sets. Building on the ideas of the degrees of openness and non-openness, Coker and Demirci [8] defined intuitionistic fuzzy topological spaces in Sostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces. Kim and Lee [9] introduced the concept of generalized fuzzy (r, s)-closed sets for intuitionistic fuzzy topological spaces in Sostak’s sense. They also investigated several properties of these sets.

Based on Levine’s concept of a generalized closed set in topological spaces [10], the concept of generalized fuzzy continuous mappings in fuzzy topological spaces was introduced by Balasubramanian and Sundaram [11]. Subsequently, El-Shafei and Zakari [12] introduced the concept of semi-generalized continuous mapping of Chang’s fuzzy topological spaces.

In this paper, we introduce four types of continuity, namely, the concepts of generalized fuzzy (r, s)-continuous, semi-generalized fuzzy (r, s)-continuous, generalized fuzzy (r, s)-semi-continuous, and generalized fuzzy (r, s)-irresolute mappings, based on the notion of generalized fuzzy (r, s)-closed sets proposed by Kim and Lee [9]. We analyze the properties of these mappings and examine the relationships among these continuities.

For the nonstandard definitions and notation, we refer to [9, 13–17]. Let I(X) be a family of all intuitionistic fuzzy sets in X and I ⊗ I be the set of the pair (r, s) such that r, s ∈ I and r + s ≤ 1.

Definition 2.1 ([8])

Let X be a nonempty set. intuitionistic fuzzy topology in Sostak’s sense (SoIFT), on X is the mapping that satisfies the following properties:

• (1) and .

• (2) and .

• (3) and .

is said to be an intuitionistic fuzzy topological space in Sostak’s sense (SoIFTS). Also, we call a gradation of openness of A and a gradation of nonopenness of A.

Definition 2.2 ([13])

Let A be an intuitionistic fuzzy set in a SoIFTS ( ) and (r, s) ∈ I ⊗ I. Then A is said to be

• (1) fuzzy (r, s)-semiopen if there is a fuzzy (r, s)-open set B in X such that B ⊆ A ⊆ cl(B, r, s),

• (2) fuzzy (r, s)-semiclosed if there is a fuzzy (r, s)-closed set B in X such that int(B, r, s) ⊆ A ⊆ B.

Theorem 2.3 ([13])

Let A be an intuitionistic fuzzy set in a SoIFTS ( ) and (r, s) ∈ I ⊗ I. Then the following statements are equivalent.

• (1) A is a fuzzy (r, s)-semiopen set.

• (2) A^c is a fuzzy (r, s)-semiclosed set.

• (3) cl(int(A, r, s), r, s) ⊇ A.

• (4) int(cl(A^c, r, s), r, s) ⊆ A^c.

Definition 2.4 ([13, 15])

Let be a mapping from a SoIFTSX for SoIFTS Y and (r, s) ∈ I⊗I. Then f is termed

• (1) a fuzzy (r, s)-continuous mapping if f⁻¹(B) is a fuzzy (r, s)-open set in X for each fuzzy (r, s)-open set B in Y,

• (2) a fuzzy (r, s)-semicontinuous mapping if f⁻¹(B) is a fuzzy (r, s)-semiopen set in X for each fuzzy (r, s)-open set B in Y.

Definition 2.5 ([9])

Let A be an intuitionistic fuzzy set in a SoIFTS ( ) and (r, s) ∈ I ⊗ I. Then A is said to be generalized fuzzy (r, s)-closed if cl(A, r, s) ⊆ B when A ⊆ B and B is fuzzy (r, s)-open. The complement of a generalized fuzzy (r, s)-closed set is termed generalized fuzzy (r, s)-open.

Definition 2.6 ([9])

Let ( ) be a SoIFTS. For each (r, s) ∈ I ⊗ I and for each A ∈ I(X), the generalized fuzzy (r, s)-closure is defined as

Definition 2.7 ([9])

Let A be an intuitionistic fuzzy set in a SoIFTS ( ) and (r, s) ∈ I ⊗ I. Then A is said to be semi-generalized fuzzy (r, s)-closed if scl(A, r, s) ⊆ B whenever A ⊆ B and B is fuzzy (r, s)-semiopen. The complement of a semi-generalized fuzzy (r, s)-closed set is termed semi-generalized fuzzy (r, s)-open.

Definition 2.8 ([9])

Let ( ) be a SoIFTS. For each (r, s) ∈ I ⊗ I and for each A ∈ I(X), the semi-generalized fuzzy (r, s)-closure is defined by

Definition 2.9 ([9])

Let A be an intuitionistic fuzzy set in a SoIFTS ( ) and (r, s) ∈ I ⊗ I. Then A is said to be generalized fuzzy (r, s)-semiclosed if scl(A, r, s) ⊆ B whenever A ⊆ B and B is fuzzy (r, s)open. The complement of a generalized fuzzy (r, s)-semiclosed set is termed generalized fuzzy (r, s)-semiopen.

Definition 2.10 ([9])

Let ( ) be a SoIFTS. For each (r, s) ∈ I ⊗ I and for each A ∈ I(X), the generalized fuzzy (r, s)-semiclosure is defined as

Definition 2.11 ([11])

Let be a mapping from a fuzzy topological space X to a fuzzy topological space Y. Then, f is termed a generalized fuzzy continuous mapping if f⁻¹(η) is a generalized fuzzy closed set in X for each fuzzy closed set η in Y.

Based on the notion of generalized fuzzy (r, s)-closed sets proposed by Kim and Lee [9], we define the generalized fuzzy (r, s)-continuous mappings on intuitionistic fuzzy topological spaces in Sostak’s sense and establish certain of their properties.

Definition 3.1

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗I. Then, f is termed a generalized fuzzy (r, s)-continuous mapping if f⁻¹(B) is a generalized fuzzy (r, s)-closed set in X for each fuzzy (r, s)-closed set. B in Y.

Remark 3.2

It is clear that every fuzzy (r, s)-continuous mapping is generalized fuzzy (r, s)-continuous. However, the following example shows that the converse need not be true.

Example 3.3

Let X = {x, y, z}, Y = {a, b} and let A₁ and A₂ be intuitionistic fuzzy sets in X and Y, respectively, defined as

and

Define and , by

and

Then, and are SoIFTs on X and Y. respectively. Consider a mapping defined by f(x) = f(z) = b, f(y) = a. Since f⁻¹(A₂) is not fuzzy (12,13)-open in X. f is not fuzzy (12,13)-continuous. Note that f-1(A2c)(x)=(0.1,0.5), f-1(A2c)(y)=(0.5,0.1), f-1(A2c)(z)=(0.1,0.5). Thus, the only fuzzy (12,13)-open set in X that contains f-1(A2c) is 1, hence, cl(f-1(A2c),12,13)⊆1_. Therefore, f-1(A2c) is a generalized fuzzy (12,13)-colsed set in X. Hence, f is generalized fuzzy (12,13)-continuous.

Theorem 3.4

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. Subsequently, f is generalized fuzzy (r, s)-continuous if and only if f⁻¹(B) is generalized fuzzy (r, s)-open in X for each fuzzy (r, s)-open set B in Y.

Theorem 3.5

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. If f is generalized fuzzy (r, s)-continuous, then f(gcl(A, r, s)) ⊆ cl(f(A), r, s), where A is any intuitionistic fuzzy set in X.

Example 3.6

Let X = {x, y, z} and let A₁ be an intuitionistic fuzzy set in X defined as

We define by

Then, is a SoIFT on X. Consider a mapping defined by f(x) = y, f(y) = x, f(z) = z. Since cl(f(A1),12,13)=1_, we have g(gcl(A1,12,13))⊆cl(f(A1),12,13). However, we note that f-1(A1c)(x)=(0.4, 0.3), f-1(A1c)(y)=(0.3, 0.4) and f-1(A1c)(z)=(0.3, 0.4). So, f-1 (A1c)⊆A1, and A₁ is a fuzzy (12,13)-open set in X. But cl(f-1(A2c),12,13)=1_⊈A1. Thus, f-1(A1c) is not generalized fuzzy (12,13)-colsed in (X, T). Therefore, f is not generalized fuzzy (12,13)-continuous.

Levine [10] introduced T12space, in which every generalized closed set is closed. The property of T12 lies strictly between T₁ and T₀. We provide a similar definition of intuitionistic fuzzy topological spaces as follows:

Definition 3.7

Let ( ) be a SoIFTS and (r, s) ∈ I ⊗ I. Subsequently, ( ) is said to be fuzzy (r,s)-T12. if every generalized fuzzy (r, s)-closed set in X is fuzzy (r, s)-closed in X.

Theorem 3.8

Let and be mappings, Y be fuzzy (r,s)-T12, and (r, s) ∈ I ⊗ I. If f and g are generalized fuzzy (r, s)-continuous, then g ∘ f is generalized fuzzy (r, s)-continuous.

Example 3.9

Let X = {x, y, z} and let A₁, A₂, A₃, and A₄ be intuitionistic fuzzy sets in X defined as

and

Define and by

and

Then, , and are SoIFTs on X. Moreover, ( ) is not fuzzy (12,13)-T12. Define as f(x) = f(z) = z, f (y) = y and let be the identity mapping. Thus, it is easy to see that f and g are generalized fuzzy (12,13)-continuous. However, g ∘ f is not. generalized fuzzy (12,13)-continuous because (g∘f)-1(A4c)=f-1(g-1(A4c))=f-1(A4c) is not generalized fuzzy (12,13)-closed in ( ).

Definition 4.1

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I⊗I. Subsequently, f is termed a semi-generalized fuzzy (r, s)-continuous mapping if f⁻¹(B) is a semi-generalized fuzzy (r, s)-closed set in X for each fuzzy (r, s)-closed set B in Y.

Theorem 4.2

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. If f is fuzzy (r, s)-semicontinuous, then f is semi-generalized fuzzy (r, s)-continuous.

Example 4.3

Let X = {x, y} and let A₁ and A₂ be intuitionistic fuzzy sets in X defined as

and

Define and , by

and

Then, and are SoIFTs on X. Consider a mapping defined by f(x) = x, f(y) = y. Then it is easy to see that f is semi-generalized fuzzy (12,13)continuous. But f is not fuzzy (12,13)-semi continuous because f⁻¹(A₂) = A₂ is not fuzzy (12,13)-semiopen in ( ).

Theorem 4.4

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. Then, f is semi-generalized fuzzy (r, s)-continuous if and only if f⁻¹(B) is semi-generalized fuzzy (r, s)-open set in X for each fuzzy (r, s)-open set, B in Y.

Theorem 4.5

Let be a semi-generalized fuzzy (r, s)-continuous mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. Then the following statements hold:

• (1) f(sgcl(A, r, s)) ⊆ cl(f(A), r, s) for each intuitionistic fuzzy set A in X.

• (2) sgcl(f⁻¹(B), r, s) ⊆ f⁻¹(cl(B, r, s)) for each intuitionistic fuzzy set B in Y.

Theorem 4.6

Let and be mapping, and (r, s) ∈ I ⊗ I. If f is semi-generalized fuzzy (r, s)-continuous and g is fuzzy (r, s)-continuous, then g ∘ f is semi-generalized fuzzy (r, s)-continuous.

Definition 4.7

Let ( ) be a SoIFTS and (r, s) ∈ I ⊗ I. Then, ( ) is said to be fuzzy(r,s)-semi-T12 if every semi-generalized fuzzy (r, s)-closed set is fuzzy (r, s)-semiclosed.

Theorem 4.8

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I⊗I. If ( ) is fuzzy, (r,s)-semi-T12, then f is semi-generalized fuzzy (r, s)-continuous if and only if f is fuzzy (r, s)-semicontinuous.

Proof. Let B be a fuzzy (r, s)-closed set in Y. Since f is semi-generalized fuzzy (r, s)-continuous, f⁻¹(B) is Semi-generalized fuzzy (r, s)-closed. By assumption, f⁻¹(B) is fuzzy (r, s)-semiclosed. Thus, f is fuzzy (r, s)-semicontinuous.

The converse follows from Theorem 4.2.

Definition 5.1

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗I. Then, f is termed a generalized fuzzy (r, s)-semicontinuous mapping if f⁻¹(B) is a generalized fuzzy (r, s)-semiclosed set in X for each fuzzy (r, s)-closed set B in Y.

Theorem 5.2

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I⊗I. If f is a semi-generalized fuzzy (rs)-continuous, then f is generalized fuzzy (r, s)-semicontinuous.

Example 5.3

Let X = {x, y} and let A₁ and A₂ be intuitionistic fuzzy sets in X defined as

and A2(x)=(0.2,0.7),A2(y)=(0,1).

Define and , by

and

Then, and are SoIFTs on X. Consider a mapping defined by f(x) = x, f(y) = y. Then f is generalized fuzzy (12,13)-semicontinuous. But f is not semi-generalized fuzzy (12,13)-continuous. For A2c⊆A2c and A2c is fuzzy (12,13)-semiopen but scl (A2c,12,13)=1_⊈A2c in ( ).

Theorem 5.4

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. If f is generalized fuzzy (r, s)-continuous, then f is generalized fuzzy (r, s)-semicontinuous.

Example 5.5

Let X = {x} and let A₁, A₂ and A₃ be intuitionistic fuzzy sets in X defined as

Define and , by

and

Then, and are SoIFTs on X. Consider a mapping is defined by f(x) = x.

We consider only the non-trival fuzzy (12,13)-closed set A3c in ( ). For the only non-trivial fuzzy (12,13)-open superset A₂ of A3c in ( ), we have scl (A3c,12,13)=A1c∩A2c∩A2=A2⊆A2 . Thus, the inverse image f-1 (A3c)=A3c is generalized fuzzy (12,13)-semiclosed in ( ). Hence, f is generalized fuzzy (12,13)-semi continuous.

We now consider the same fuzzy (12,13)-closedset A3c in ( ). For the fuzzy (12,13)-open superset A₂ of A3c in ( ), cl (A3c,12,13)=A2c⊈A2. Thus the inverse image f-1(A3c)=A3c is not generalized fuzzy (12,13)-closed in ( ). Hence f is not generalized. fuzzy (12,13)-continuous.

Theorem 5.6

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. Subsequently, f is generalized fuzzy (r, s)-semicontinuous if and only if f⁻¹(B) is a generalized fuzzy (r, s)-semiopen set in X for each fuzzy (r, s)-open set B in Y.

Theorem 5.7

Let be a generalized fuzzy (r, s)-semicontinuous mapping from aSoIFTS X to aSoIFTS Y and (r, s) ∈ I ⊗ I. Thus, the following statements hold:

• (1) f(gscl(A, r, s)) ⊆ cl(f(A), r, s) for each intuitionistic fuzzy set A in X.

• (2) gscl(f⁻¹(B), r, s)) ⊆ f⁻¹(cl(B, r, s)) for each intuitionistic fuzzy set B in Y.

Theorem 5.8

Let and be mapping, and (r, s) ∈ I ⊗I. If f is generalized fuzzy (r, s)-semicontinuous and g is fuzzy (r, s)-continuous. Then, g ∘ f is generalized fuzzy (r, s)-semicontinuous.

Definition 6.1

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗I. Then, f is said to be generalized fuzzy (r, s)-irresolute if f⁻¹(B) is a generalized fuzzy (r, s)-closed set in X for each generalized fuzzy (r, s)-closed set B in Y.

Theorem 6.2

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. Subsequently, f is generalized fuzzy (r, s)-irresolute if and only if f⁻¹(B) is generalized fuzzy (r, s)-open in X for each generalized fuzzy (r, s)-open set B in Y.

Theorem 6.3

Let be a mapping from a SoIFTS X to a SoIFTS Y and (r, s) ∈ I ⊗ I. If f is generalized fuzzy (r, s)-irresolute, then f is generalized fuzzy (r, s)-continuous.

Example 6.4

Let X = {x, y, z} and A₁, A₂, and A₃ be intuitionistic fuzzy sets in X defined as

and

Define and , by

and

Then, and are SoIFTs on X. Consider a mapping defined by f(x) = x, f(y) = y, and f(z) = x. Clearly, f is generalized fuzzy (12,13)-continuous. However, f is not generalized fuzzy (12,13)-irresolute, because A₃ is generalized fuzzy (12,13)-closed in ( ) and f⁻¹(A₃) = A₃ is not generalized fuzzy (12,13)-closed in ( ).

Theorem 6.5

Let and be mappings and (r, s) ∈ I ⊗I. If f is generalized fuzzy (r, s)-irresolute and g is generalized fuzzy (r, s)-continuous, then g ∘ f is generalized fuzzy (r, s)-continuous.

We introduce four types of continuity, i.e., the concepts of generalized fuzzy (r, s)-continuous, semi-generalized fuzzy (r, s)-continuous, generalized fuzzy (r, s)-semicontinuous, and generalized fuzzy (r, s)-irresolute mappings and investigated several of their properties. These continuities are either strong or weak versions of the traditional continuities. The relationships among these continuities were also discussed.