Since Zadeh [1] introduced the notion of fuzzy sets, several researchers have attempted to generalize fuzzy sets using various approaches.

One approach has been to use the concept of intuitionistic fuzzy sets introduced by Atanassov [2]. Unlike fuzzy sets, an intuitionistic fuzzy set provides both a membership degree and a non-membership degree. Many important concepts in general topology have been generalized to intuitionistic fuzzy settings.

The other main approach is to utilize the concept of rough sets introduced by Pawlak [3]. The rough set theory proposes a mathematical approach to imperfect knowledge, which is expressed by the boundary region of a set. The rough set concept can be represented by topological approximations, that is, interior and closure. Zhou et al. [4,5] proposed lower and upper approximations of intuitionistic fuzzy sets. Many researchers, including us, have examined the important properties of intuitionistic fuzzy approximation operators [6–8].

Many attempts have been made with the objective of combining fuzziness and roughness. Dubois and Prade [9] and Nanda and Majumda [10] introduced and discussed the concept of fuzzy rough sets. Coker [11] demonstrated that fuzzy rough sets, as proposed by Nanda and Majumdar [10], are intuitionistic L-fuzzy sets developed by Atanassov [2]. Chakrabarty et al. [12] proposed a fuzziness measure in rough sets. By combining the concepts of intuitionistic fuzzy sets and fuzzy rough sets, Samanta and Mondal [13] proposed the idea of intuitionistic fuzzy rough sets. The topology of intuitionistic fuzzy rough sets was accordingly introduced [14,15]. Bashir et al. [16] studied the topological properties of intuitionistic fuzzy rough sets under different conditions like serial, strongly serial, and left continuity.

The concept of intuitionistic fuzzy rough sets has some advantages with regard to decision-making. Zhan and Sun [17] discussed the rough and precision degrees of covering-based intuitionistic fuzzy rough set models. They introduced an intuitionistic fuzzy rough methodology to the multi-attribute decision-making problem, which is more effective than the previous models. Shanthi [18] developed a decision-making method based on the composition of intuitionistic fuzzy rough matrices on a finite universe.

In light of mathematical theory, however, the properties of the “old” intuitionistic fuzzy rough sets are extremely complicated and inadequate in terms of the extension of intuitionistic properties. Hence, remedying this flaw is critical for expanding related theories. To overcome this weakness, we introduced a new definition of fuzzy rough sets and intuitionistic fuzzy rough sets [19].

In this paper, we introduce a topology for redefined intuitionistic fuzzy rough sets and investigate the basic properties of their subspaces, transition spaces, and continuous functions. In addition, we study the categorical relation between the category of fuzzy rough sets and the category of intuitionistic fuzzy rough sets. The results obtained from this new definition are different from those obtained in previous studies. Some of our results are similar to those of Hazra et al. [15]. However, our results are consistent with the theory of intuitionistic fuzzy sets. The results help us base decision-making on a more solid foundation.

In [10], the definition of fuzzy rough sets was introduced. The paper stated “We shall consider (V, ) to be a rough universe where V is a nonempty set and is a Boolean subalgebra of the Boolean algebra of all subsets of V. In addition, consider a rough set. with X_L ⊆ X_U. A fuzzy rough set of X is an object of the form

where A_L and A_U are characterized by a pair of maps A_L : X_L → L and A_U : X_U → L with A_L(x) ≤ A_U(x) for all x ∈ X_L, where (L, ≤) is a fuzzy lattice.”

Furthermore, the complement Ā of a fuzzy rough set A = (A_L, A_U) is defined by (Ā)_L(x) = (A_U>L)′(x), ∀x ∈ X_L, and (Ā)_U(x) = (A_L<U)′(x), ∀x ∈ X_U,

where A_U>L(x) = A_U(x), ∀x ∈ X_L and

Unfortunately, if we follow this definition, the double complement of fuzzy rough set A is different from A. This is because X_L and X_U could be different. The property that the double complement of a set becomes the set itself is essential in expanding related theories. To avoid this, we introduced a new definition of fuzzy rough sets and intuitionistic fuzzy rough sets by weakening the condition of the old definition. We recall some definitions and the results of our previous study.

Definition 2.1 ([19])

Let X be an underlying set and (L, ≤) a fuzzy lattice. A fuzzy rough set of X is an object of the form

where A_L and A_U are defined by a pair of maps A_L : X → L and A_U : X → L with A_L(x) ≤ A_U(x) for all x ∈ X. Furthermore, 0 = (0_L, 0_U) is the null fuzzy rough set, and 1 = (1_L, 1_U) is the entire fuzzy rough set of X.

The complement Ā = ((Ā)_L, (Ā)_U) of a fuzzy rough set A = (A_L, A_U) of X is defined by (Ā)_L(x) = (A_U(x))′ and (Ā)_U(x) = (A_L(x))′ for all x ∈ X, where a′ denotes the complement for a ∈ L.

Theorem 2.2

For any fuzzy rough set A = (A_L, A_U) of X, (A¯)¯=A.

Definition 2.3. [19]

If A = (A_L, A_U) and B = (B_L, B_U) are two fuzzy rough sets of X with B ⊆ Ā, then the ordered pair (A, B) is called an intuitionistic fuzzy rough set (briefly, IF rough set) of X. Condition B ⊆ Ā is called the intuitionistic condition. The complement of the IF rough set P = (A, B) of X is defined by P̄ = (B, A).

An IF rough set is an intuitive version of a fuzzy rough set. An IF rough set has a membership degree and a non-membership degree comprising fuzzy rough sets. Therefore, this concept is effective in dealing with systems that have a rough membership degree and rough non-membership degree.

We denote by IFRS(X) the collection of all IF rough sets of X. Usually, we use letters X, Y, Z, … to denote sets, letters A, B, C, … to denote fuzzy rough sets, and bold letters P, Q, R, … to denote IF rough sets.

The IF rough sets 0 = (0, 1) = ((0_L, 0_U), (1_L, 1_U)), and 1 = (1, 0) = ((1_L, 1_U), (0_L, 0_U)) are called the null IF rough set and the whole IF rough set of X, respectively. Clearly, 0̄ = 1 and 1̄ = 0. If we identify the crisp set X with Χ_X = (Χ_X, 0) = (1, 0) = ((1_L, 1_U), (0_L, 0_U)) = 1_X, then X is also an IF rough set of X.

Definition 2.4 ([19])

Let f : X → Y be a map. Let P = (A, B) = ((A_L, A_U), (B_L, B_U)) be an IF rough set of X. We define an image of P under f as follows:

where f(A) = (f(A_L), f(A_U)), and f(B¯)¯=((f(B¯)¯)L,(f(B¯)¯)U) is defined as (f(B¯)¯)L=f((B¯)L)¯ and (f(B¯)¯)U=f((B¯)U)¯.

Subsequently, f(P) satisfies the intuitionistic condition (Remark 3.9 of [19]); hence, f(P) is an IF rough set. For any IF rough set P = (A, B) of X and any map f : X → Y, the image f(P) = f((A, B)) of P under f is expressed as follows: For any y ∈ Y,

Definition 2.5 ([19])

Let f : X → Y be a mapping and Q = (C, D) = ((C_L, C_U), (D_L, D_U)) be an IF rough set of Y. We define an inverse image of Q under f as:

where f⁻¹(C) = (f⁻¹(C_L), f⁻¹(C_U)) and f⁻¹(D) = (f⁻¹(D_L), f⁻¹(D_U)).

Then, the inverse image of an IF rough set under f is also an IF rough set.

For the other properties of IF rough sets, refer to [19].

One of the benefits of our new definition is the ability to construct a category of rough sets and a category of IF rough sets, where the double negation of a fuzzy rough set becomes itself. With the old definition of [15], constructing a category is difficult.

In this section, we study the connection between two concepts, that is, IF rough sets and fuzzy rough sets.

Definition 3.1

Let A be a fuzzy rough set of X and C a fuzzy rough set of Y. A function f : X → Y is called a function from A to C, denoted by f : A → C, if f(A) ⊆ C.

Definition 3.2

Let P = (A, B) be an IF rough set of X and Q = (C, D) an IF rough set of Y. A function f : X → Y is called a function from P to Q, denoted by f : P → Q, if f(P) ⊆ Q.

Now, we are ready to define functors between two categories. Let FRS be the category of all fuzzy rough sets and functions on fuzzy rough sets. Let IFRS is the category of all IF rough sets and functions on IF rough sets.

Theorem 3.3

Define F₁ : IFRS → FRS by

Then F₁ is a functor from IFRS to FRS.

Theorem 3.4

Define F₂ : IFRS → FRS by

Then, F₂ is a functor from IFRS to FRS.

Theorem 3.5

Define G₁ : FRS → IFRS by

Then, G₁ is a functor from FRS to IFRS.

Theorem 3.6

Define G₂ : FRS → IFRS by

Then, G₂ is a functor from FRS to IFRS.

As in the following two theorems, we have two adjointnesses between the above functors.

Theorem 3.7

The functor F₁ : IFRS → FRS is a right adjoint of functor G₁ : FRS → IFRS.

Theorem 3.8

The functor F₂ : IFRS → FRS is a right adjoint of functor G₂ : FRS → IFRS.

A basic approach for studying the application of a new mathematical system is topology because the topological structure explains the convergence and neighborhoodness of a system. In this section, we introduce the topology of IF rough sets.

Definition 4.1

Let X be a nonempty set and be a family of IF rough sets of X such that

• ,

• for all ,

• for all i ∈ Δ implies .

Then, is called a topology of IF rough sets of X, and the pair (X, ) is called a topological space of IF rough sets of X. Every member of is called an open IF rough set. An IF rough set Q is said to be closed if . Let ℱ denote the collection of all closed IF rough sets of (X, ). If , then is a topology of IF rough sets of X. This topology is called a indiscrete topology. The discrete topology of IF rough sets of X contains all IF rough sets of X.

Theorem 4.2

The collection of ℱ of all closed IF rough sets of (X, ) satisfies the following properties:

• 0, 1_X ∈ ℱ.

• P ∪ Q ∈ ℱ, for all P, Q ∈ ℱ.

• P_i ∈ ℱ, i ∈ Δ implies ⋂_i∈ΔP_i ∈ ℱ.

Definition 4.3

Let P be an IF rough set of X. The interior of P in (X, ) is the union of all open IF rough sets in (X, ) contained in P and is denoted by .

Definition 4.4

Let P be an IF rough set of X. The closure of P in (X, ) is the intersection of all closed IF rough sets in (X, ) containing P and is denoted by .

is the smallest closed IF rough set containing P, and P is closed if and only if . In addition, is the largest open IF rough set contained in P, and P is open if and only if . We have the following properties directly from the definitions.

Theorem 4.5

• .

• .

• P ⊆ Q implies .

• .

• .

Definition 4.6

Let (X, ) and (Y, ) be two topological spaces of the IF rough sets. And let f : X → Y be a mapping. Then, is said to be IFR continuous if for all .

Let (X, ), (Y, ), and (Z, ) be the topological spaces of the IF rough sets. If , and are IFR continuous, then is clearly IFR continuous.

The continuous function is characterized by closed sets and closure as follows.

Theorem 4.7

The following statements are equivalent:

• is IFR continuous.

• f⁻¹(Q) is closed IF rough set in (X, ), for any closed IF rough set Q in (Y, ).

• , for any IF rough set P in (X, ).

Let Y be a crisp subset of X. We denote the IF rough set (Χ_Y , 1 − Χ_Y) = (((Χ_Y )_L, (Χ_Y)_U), (1 − Χ_Y )_L, (1 − Χ_Y )_U)) by Χ_Y or 1_Y, and the IF rough set χY¯=(1-χY,χY)=((1-χY)L,(1-χY)U),((χY)L,(χY)U)) by 0_Y. If we take Y = X, then 1_Y = 1 and 0_Y = 0. Clearly, 0_Y is a complement of 1_Y and vice versa, as an IF rough set.

Theorem 5.1

Let (X, ) be a topological space of IF rough sets of X. Let Y be a crisp subset of X. Then, the family of IF rough sets of Y satisfies the axioms of Definition 4.1.

This topology is called the subspace topology of (X, ) on Y. Every member of is called an open IF rough set of (Y, ). Clearly, and .

If , then Q̄_Y is called a closed IF rough set of (Y, ), where Q̄_Y = Q̄ ∩ 1_Y. Let

Then, ℱ_Y is closed under arbitrary intersections and finite unions. Since , we have 0=1Y¯∩1Y∈ℱY. In addition, 1_Y = 0̄ ∩ 1_Y ∈ ℱ_Y.

Theorem 5.2

Let (X, ) be a topological space of IF rough sets of X. Let P be an IF rough set of Y with Y ⊆ X. Then cl_Y (P) = cl_X(P) ∩ 1_Y.

Theorem 5.3

If is continuous and Z ⊆ X, then is continuous.

Theorem 5.4

If is continuous, then is continuous.

Hazra et al. [15] defined the subspace topology, but it is, in fact, a new concept different from the subspace topology. Consequently, we suggest a new terminology “transition topology.” Insofar as we take the characteristic function of a crisp set, the transition topology and the subspace topology are the same.

Definition 6.1

Let P be an IF rough set. A topology on P is a collection of IF rough subsets of P such that the following are true:

• .

• If , then .

• If for all i ∈ Δ, then .

Theorem 6.2

Let be a topology of IF rough sets of X and P ∈ IFRS(X). Then, is clearly a topology on P, which is called a transition topology on P of (X, ). The pair (P, ) is called a transition space of (X, ).

If we take P = Χ_E for some crisp subset E of X, then the transition topology becomes the subspace topology on E of (X, ).

Every member of is called an open IF rough set of (P, ). If , then Q̄_P is called a closed IF rough set of (P, ), where Q̄_P = P ∩ Q̄. Let

Subsequently, ℱ_P is a family of closed IF rough sets in (P, ).

Theorem 6.3

ℱ_P is closed under arbitrary intersections and finite unions.

Remark 6.4

Let . Then, S̄_P = P ∩ S̄ ∈ ℱ_P. Now,

because S ⊆ P. As , (S¯P)¯P=(P∩P¯)∪(P∩R) for some . Thus, (S¯P)¯P=P∩(P¯∪R). Hence, (S¯P)¯P does not necessarily belong to . However,

By the above remark, we can give the following definition.

Definition 6.5

The collection

forms a topology of IF rough sets on P. This is called the second transition topology of (X, ) on P. The pair (P, ) is called the second transition space of (X, ).

If we take P = Χ_E for some crisp subset E of X, then the second transition topology becomes the subspace topology on E of (X, ).

ℱ_P is also a family of closed IF rough sets in (P, ). Thus, there exist two topologies of IF rough sets on P. is the transition topology of (X, ) on P, and is another transition topology of (X, ) on P. We briefly refer to and first and second topologies, respectively, on P if there is no confusion about the topological space (X, ) of IF rough sets.

Remark 6.6

In general, and are different. For P ∈ IFRS(X), let and be the first and second topologies on P, that is, , and

If is an indiscrete topology of IF rough sets of X, then and . Thus, .

If is a discrete topology, then . Thus, .

Remark 6.7

Let R ∈ IFRS(X) such that R ⊆ P. The closures of R in (P, ) and (P, ) are denoted by and , respectively. Because the closed IF rough sets in (P, ) and (P, ) are the same, . Furthermore,

Thus, .

Following example shows that, in general

Example 6.8

Let X = [0, 1]. Let C = (C_L, C_U) and D = (D_L, D_U) be the fuzzy rough sets of X, where

Then, Q = (C, D) is an IF rough set of X. Let . Thus, (X, ) is a topological space of IF rough sets.

Let A = (A_L, A_U) and B = (B_L, B_U) be the fuzzy rough sets of X, where

Then, P = (A, B) is an IF rough set of X. Define . Then, is a first topology on P.

Let M = (M_L, M_U) and N = (N_L, N_U) be the fuzzy rough sets of X, where

Then, R = (M, N) is an IF rough set of X such that R ⊆ P.

Now, clT1R=P∩(∩{(P∩G)¯∣(P∩G)¯⊇R;G∈T})=P∩(∩{P¯∪G¯∣P¯∪G¯⊇R,G∈T})..

because A ⊈ N. We then have P̄ ∪ Q̄ = (B ∪ D, A ∩ C) ⊇ R. Thus, . If we write this set as (F, G), then

We have . Thus, 0̄ = 1 ⊇ R, 1̄ = 0 ⊉ R, Q̄ ⊉ R, because C ⊈ N. Thus, . Therefore,

Clearly, (F, G) ⊆ (A, B) and (F, G) ≠ (A, B). Thus, is a proper subset of .

In this paper, we introduced a topology for the redefined intuitionistic fuzzy rough sets and investigated the basic properties of their subspaces and continuous functions. We also introduced the concepts of the first and second transition spaces. We confirmed that these two spaces are different but share the same closed sets. In addition, we obtained the adjointness between the categories of fuzzy rough sets and intuitionistic fuzzy rough sets. All results obtained from this new definition are different from those of previous studies. The new approach of intuitionistic fuzzy rough sets enables us to manipulate them more simply and easily. In the following study, we will focus on expanding the related theories.