On the Characterization of Strong (I, J)-RW Closed Sets in Bitopological Spaces

Bitopological spaces consist of a non-empty set X and two arbitrary topologies T1 and T2. This concept was first introduced by Kelly in 1963 and has since become a vital tool in modern topology. While standard topological spaces look at one way of organizing a set, bitopology allows us to look at two different perspectives simultaneously. Recent developments in Generalized Closed Sets have led to the discovery of regular-open and omega-closed structures. This paper merges these ideas to define Strong (i,j)-Rw closed sets, providing a stronger condition for set-theoretic closure in environments where data is observed through multiple sensors or perspectives.

3. Preliminaries and Definitions

Let (X, T1, T2) be a bitopological space.

Definition 3.1: For any subset A of X, cl-i(A) denotes the closure of A with respect to the topology Ti.

Definition 3.2: For any subset A of X, int-i(A) denotes the interior of A with respect to the topology Ti.

Definition 3.3: A subset A is called (i,j)-regular open if A = int-i(cl-j(A)).

Definition 3.4: A subset A is called (i,j)-g-closed if cl-j(A) is a subset of U whenever A is a subset of U and U is Ti-open.

Definition 3.5: A subset A is (i,j)-omega-closed if cl-j(A) is contained in U whenever A is contained in U and U is Ti-open.

4. Strong (I, J)-RW Closed Sets

Definition 4.1: A subset A of a bitopological space (X, T1, T2) is called a Strong (i,j)-Rw closed set if cl-j(A) is a subset of U whenever A is a subset of U and U is (i,j)-Rw open in X.

Remark 4.1: Here, i and j are distinct elements of the set {1, 2}. If i = 1, then j = 2, and vice-versa.

5. Theorems and Rigorous Proofs

Theorem 5.1: Every Ti-closed set is a Strong (i,j)-Rw closed set.

Proof: Let A be a Ti-closed set in X. This implies cl-i(A) = A. Let U be any (i,j)-Rw open set such that A is a subset of U. Since A is closed, its closure cl-j(A) is also equal to A. Therefore, cl-j(A) is a subset of U. This confirms that A is Strong (i,j)-Rw closed.

Theorem 5.2: If A is Strong (i,j)-Rw closed and A is a subset of B, and B is a subset of cl-j(A), then B is also a Strong (i,j)-Rw closed set.

Proof: Let U be an (i,j)-Rw open set such that B is a subset of U. Since A is a subset of B, it follows that A is also a subset of U. Because A is Strong (i,j)-Rw closed, we know cl-j(A) is a subset of U. Since B is a subset of cl-j(A), the closure of B, cl-j(B), must also be a subset of cl-j(cl-j(A)), which is just cl-j(A). Thus, cl-j(B) is a subset of U.

Theorem 5.3: The union of two Strong (i,j)-Rw closed sets is also a Strong (i,j)-Rw closed set.

Proof: Let A and B be two Strong (i,j)-Rw closed sets. Suppose the union (A union B) is a subset of U, where U is (i,j)-Rw open. This means A is a subset of U and B is a subset of U. Since A and B are Strong (i,j)-Rw closed, cl-j(A) is a subset of U and cl-j(B) is a subset of U. Therefore, cl-j(A union B), which is (cl-j(A) union cl-j(B)), must also be a subset of U.

6. Separation Axioms in Strong (I, J)-Rw Spaces

Definition 6.1: A bitopological space (X, T1, T2) is said to be Pairwise Strong (i,j)-Rw-T-half if every Strong (i,j)-Rw closed set is Ti-closed.

Theorem 6.1: If (X, T1, T2) is a Pairwise Strong (i,j)-Rw-T-half space, then the class of Ti-closed sets and Strong (i,j)-Rw closed sets coincide.

Proof: By standard definition, every Ti-closed set is Strong (i,j)-Rw closed. By the definition of T-half spaces, every Strong (i,j)-Rw closed set is Ti-closed. Thus, the two classes are identical. This property is vital for simplifying complex bitopological structures.

7. Strong (I, J)-Rw Continuity and Homeomorphisms

Definition 7.1: A function f from (X, T1, T2) to (Y, S1, S2) is called Strong (i,j)-Rw continuous if the inverse image of every Si-closed set in Y is a Strong (i,j)-Rw closed set in X.

Definition 7.2: A bijective map f is called a Strong (i,j)-Rw Homeomorphism if f is Strong (i,j)-Rw continuous and the inverse map f-inverse is also Strong (i,j)-Rw continuous.

8. Neutrosophic Mathematical Model

In a Neutrosophic set, each element x is represented by three independent values: Truth-membership (T), Indeterminacy-membership (I), and Falsity-membership (F).

Application Algorithm:

Step 1: Construct Topology T1 using Truth values.

Step 2: Construct Topology T2 using Falsity values.

Step 3: Identify the Strong (1,2)-Rw closed sets. These sets represent the "Stable Alternatives" where uncertainty is minimized across both perspectives.

Step 4: The alternative with the smallest Strong (i,j)-Rw closure is selected as the optimal choice.

9. Comparative Analysis

| Property | g-closed Sets | Rw-closed Sets | Strong (i,j)-Rw |

|---|---|---|---|

| Relies on Regularity | No | Yes | Highly |

| Symmetry | Asymmetric | Partial | Full Pairwise |

| Stability in MCDM | Low | Moderate | High |

CONCLUSION AND FUTURE SCOPE

We have successfully characterized the properties of Strong (i,j)-Rw closed sets. These sets provide a more restrictive and reliable tool for bitopological analysis compared to standard generalized closed sets. Future research will extend these concepts to Bitopological Nano-Structures and graph theory.

REFERENCE

1. Kelly, J. C. (1963). Bitopological Spaces. Proc. London Math. Soc.
2. Lellis Thivagar, M. (1991). Generalized Closed Sets in Bitopological Spaces.
3. Vasanthi, R. (2026). Recent Trends in Neutrosophic Applied Mathematics.