Generalized separation via γ-Sets in L-Fuzzy topological spaces

Introduction

Let S be the set of all non-negative integers. Connectedness is a fundamental concept in fuzzy topology. In 1975, Bruce Hutton [10] constructed the L-fuzzy unit interval I(L) and investigated its fuzzy topological properties, where I = [0, 1]. He proved that when L is orthocomplemented, the fuzzy open sets of I(L) correspond one-to-one with the usual open sets of I, and this correspondence preserves unions and intersections. As a consequence, I(L) possesses many fuzzy topological properties, including connectedness. However, he observed that I(L) fails to be connected when L is a chain. Later, Rodabaugh [14] showed that I(L), the L-fuzzy open unit interval (0, 1)(L), and the L-fuzzy real line R(L) are connected if 0 ∈ L b (a condition satisfied by chains), where L b = {α ∈ L: α < β and α < γ ⇒ α < β ∧ γ}. This condition is satisfied by chains. His observation can be summarized as follows: a greater degree of connectivity should be associated with a lesser degree of dis-connectivity. Furthermore, Zheng Chong-You [17] studied several properties of the L-fuzzy unit interval and introduced the concept of L-fuzzy path connectedness. He examined the relationships among different types of fuzzy connectedness, such as O-connectedness, Q-connectedness, 1-connectedness, and standard L-fuzzy path connectedness. Ajmal and Kohli [3] introduced the concepts of fuzzy connectedness and provided their characterizations. They observed that fuzzy connectedness remains invariant under fuzzy continuous mappings, highlighting its stability as a topological property. In [4] , it was further demonstrated that fuzzy 𝐶 2 -connectedness and fuzzy 𝐶 4 -connectedness are not necessarily preserved under the product of spaces. This result shows that certain types of fuzzy connectedness behave differently when forming product spaces. Turanli [15] presented several counterexamples concerning the product of fuzzy connected spaces. Moreover, he introduced the notion of fuzzy super 𝐶 𝑠 -connectedness and investigated its relationship with fuzzy 𝐶 𝑠connectedness, thereby extending the existing theory. Liu and Pu [13] introduced the concept of connectedness in fuzzy topological spaces in 1980. Their work laid the foundation for further research in this area. Subsequently, G. J. Wang [16] extended the definition proposed by Liu and Pu to the broader framework of L-fuzzy topological spaces, thereby generalizing the concept. Many researchers have since investigated different forms of connectedness [1,2,8] in L-fuzzy topological spaces, particularly in the sense defined by Chang [6] . These studies have contributed significantly to the development of fuzzy topology. Using pre-closed sets, Bai [5] introduced the notion of P-connectedness, which provided a new perspective on connectedness in L-fuzzy spaces. These concepts further enriched the theory by offering refined classifications of connectedness. Recently, El-Atik [7] investigated the notion of b-connectedness and explored its applications in general topology. His work contributed to a deeper understanding of alternative forms of connectedness and their significance in topological structures. Hanafy [9] introduced the concept of a γ-open set and defined γ-connectedness in fuzzy topological spaces. He also examined several properties of γ-open sets and studied fuzzy γ-continuity in such spaces. These investigations helped clarify the behavior of γ-related structures and enriched the theory of fuzzy topology by providing new tools for analysis. In this paper, we define the concept of γ-separated sets in L-fuzzy topological spaces using γ-open sets. This approach provides a generalized framework for studying separation in fuzzy topology. It is observed that every separated set is also γ-separated, showing that γ-separation is a broader notion. We investigate various characterizations of γ-separated sets and establish several of their important properties. These results contribute to a deeper understanding of separation axioms in L-fuzzy topological spaces.

Preliminaries

In this section, we present some basic concepts and preliminary notions that will be useful for developing our main results. Liu [12] introduced the notion of separated sets in the framework of L-fuzzy topological spaces. These sets are commonly referred to as separated L-fuzzy sets. The concept plays an important role in understanding the structural properties of L-fuzzy topological spaces. In the following, we recall and discuss some important definitions introduced by Liu [12] , which form the foundation for our subsequent study. Definition 2.1. [12] Let (X, τ) be an L-fuzzy topological space and 𝐴, 𝐵 ∈ 𝐿 𝑋 . Then 𝐴 and 𝐵 are called separated if 𝐴 ∧ 𝑐𝑙(𝐵) = 0 and 𝐵 ∧ 𝑐𝑙(𝐴) = 0. Definition 2.2. [12] Let (X, τ) be an L-fuzzy topological space and 𝐴 ∈ 𝐿 𝑋 . Then A is said to be connected if A cannot be represented as join of two separated non-null L-fuzzy sets. Definition 2.3. [9] Let (X, τ) be an L-fuzzy topological space and 𝐴 ∈ 𝐿 𝑋 . Then A is said to be γ-open

Definition 2.4. [9] Let (X, τ) be an L-fuzzy topological space and 𝐴 ∈ 𝐿 𝑋 . The fuzzy 𝛾-interior of 𝐴 (in short 𝛾-int (𝐴)) is defined as 𝛾-int (𝐴) = ⋁{𝐻: 𝐻 𝑖𝑠 𝛾 -𝑜𝑝𝑒𝑛 𝐿 -𝑓𝑢𝑧𝑧𝑦 𝑠𝑒𝑡, 𝐻 ≤ 𝐴}. The 𝛾-interior of A is the join of all γ-open subsets of 𝐿 𝑋 contained in A. The γ-interior of 𝐴 ∈ 𝐿 𝑋 is just largest γ-open subset of 𝐿 𝑋 contained in A. Note that L-fuzzy set A is γ-open iff γ-int (A) = A. Definition 2.5. [9] Let (X, τ) be an L-fuzzy topological space and 𝐴 ∈ 𝐿 𝑋 . The fuzzy 𝛾-closure of 𝐴 (in short 𝛾-cl (𝐴)) is defined as 𝛾-cl (𝐴) = ⋀{𝐺: 𝐺 𝑖𝑠 𝛾 -𝑐𝑙𝑜𝑠𝑒𝑑 𝐿 -𝑓𝑢𝑧𝑧𝑦 𝑠𝑒𝑡, 𝐴 ≤ 𝐺}. The 𝛾-closure of A is the meet of all γ-closed subsets of 𝐿 𝑋 containing A. The γ-closure of 𝐴 ∈ 𝐿 𝑋 is just smallest γ-closed subset of 𝐿 𝑋 containing A. Note that L-fuzzy set A is γ-closed iff γ-cl (A) = A.

Main Results

We are now in a position to introduce the concept of γ-separated L-fuzzy sets within the setting of L-fuzzy topological spaces. In this section, we formally define γ-separated L-fuzzy sets and explore their fundamental features. Furthermore, we prove that every separated L-fuzzy set is also γ-separated, thereby establishing a significant relationship between these two notions. In addition, we investigate several characterizations and study various important properties of γ-separated sets, which contribute to a deeper understanding of their structural behavior in L-fuzzy topological spaces. Definition 3.1. Let (X, τ) be an L-fuzzy topological space and 𝐴, 𝐵 ∈ 𝐿 𝑋 . Then 𝐴, 𝐵 are called 𝛾-separated if 𝐴 ∧ 𝛾 -𝑐𝑙(𝐵) = 0 and 𝐵 ∧ 𝛾 -𝑐𝑙(𝐴) = 0. Theorem 3.1. Every separated L-fuzzy set in L-fuzzy topological space (𝑋, 𝜏) is γseparated.

Proof. Let 𝐴 and 𝐵 are separated L-fuzzy sets in L-fts (𝑋, 𝜏). By Definition 2.1, we have 𝐴 ∧ 𝑐𝑙(𝐵) = 0 and 𝐵 ∧ 𝑐𝑙(𝐴) = 0. By using the fact 𝛾 -𝑐𝑙(𝐴) ≤ 𝑐𝑙(𝐴) for every 𝐴 ∈ 𝐿 𝑋 , we get 𝐴 ∧ 𝛾 -𝑐𝑙(𝐵) = 0 and 𝐵 ∧ 𝛾 -𝑐𝑙(𝐴) = 0. Hence by Definition 2.6, 𝐴 and 𝐵 are 𝛾-separated L-fuzzy sets in L-fts (𝑋, 𝜏). Theorem 3.2. Let (X, τ) be L-fuzzy topological space and 𝐴, 𝐵 ∈ 𝐿 𝑋 such that 𝐴 ≠ 0 𝑎𝑛𝑑 𝐵 ≠ 0. Then following statements hold;

1. If A and B are γ-separated L-fuzzy sets and 𝐴 1 , 𝐵 1 are non-null L-fuzzy sets in L-fts (X, τ) such that 𝐴 1 ≤ 𝐴 and 𝐵 1 ≤ 𝐵, then 𝐴 1 , 𝐵 1 are also γ-separated. Hence, 𝐴⋀𝐵 and 𝐴⋀𝐶 are 𝛾-separated.

Conclusion

In this paper, we have introduced and developed the concept of γ-separated sets in L-fuzzy topological spaces as a natural generalization of classical separation axioms. By employing γ-open sets, a broader and more flexible framework for separation has been established. We proved that every separated L-fuzzy set is γ-separated, which confirms that γ-separation properly extends the existing notion of separation. Various characterizations and fundamental properties of γ-separated sets were investigated, providing deeper insight into their structural behavior. Furthermore, the relationships between γ-separated sets and other fuzzy separation concepts were explored, enriching the theory of L-fuzzy topology. These results contribute significantly to the advancement of fuzzy topological studies and may serve as a foundation for further research in generalized separation structures and their applications.

Acknowledgement

AcknowledgementThe author (s) sincerely thank the referees for their valuable comments, constructive suggestions, and careful review, which greatly helped in improving the quality and clarity of this manuscript. Their insightful observations and thoughtful recommendations contributed significantly to strengthening the presentation and organization of the results. The author (s) deeply appreciate the time and effort invested by the referees in evaluating this work. Their scholarly guidance has been instrumental in enhancing the overall quality of the paper.

References

1. Ali DM. Some other types of fuzzy connectedness. Fuzzy Sets Syst. 1992;46(1):55-61.
2. Ali DM, Srivastava AK. On fuzzy connectedness. Fuzzy Sets Syst. 1988;28(2):203-208.
3. Ajmal N, Kohli JK. Connectedness in fuzzy topological spaces. Fuzzy Sets Syst. 1989;31(3):369-388.
4. Ajmal N, Kohli JK. Connectedness and local connectedness in fuzzy topological spaces and heyting algebra valued sets. Fuzzy Sets Syst. 1991;44(1):93-108.
5. Bai SZ. P-connectedness in L-topological spaces. Soochow J Math. 2003;29(1):35-42.
6. Chang CL. Fuzzy topological spaces. J Math Anal Appl. 1968;24(1):182-190. (Note: Corrected year from 1986 to 1968 based on original publication).
7. El-Atik AA, Abu Donia HM, Salama AS. On b-connectedness and b-disconnectedness and their applications. J Egypt Math Soc. 2013;21(2):63-67.
8. Ghosh B. Semi-continuous and semi-closed mappings and semi-connectedness in fuzzy setting. Fuzzy Sets Syst. 1990;35(3):345-355.
9. Hanafy IM. Fuzzy γ-open sets and fuzzy γ-continuity. J Fuzzy Math. 1999;7:419-430.
10. Hutton B. Normality in fuzzy topological spaces. J Math Anal Appl. 1975;50(1):74-79.
11. Li SP, Fang Z, Zhao J. $\theta$-connectedness in L-topological spaces. Iran J Fuzzy Syst. 2005;2(1):29-36.
12. Liu YM, Luo MK. Fuzzy topology. Vol. 9, Advances in Fuzzy Systems: Applications and Theory. River Edge (NJ): World Scientific; 1997.
13. Pu PM, Liu YM. Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence. J Math Anal Appl. 1980;76(2):571-599.
14. Rodabaugh SE. Connectivity and the L-fuzzy unit interval. Rocky Mt J Math. 1982;12(1):113-121.
15. Turanli N, Coker D. Fuzzy connectedness in intuitionistic fuzzy topological spaces. Fuzzy Sets Syst. 2000;116(3):369-375.
16. Wang GJ. Theory of L-fuzzy topological space. Xi'an: Shaanxi Normal University Publishers; 1988.
17. Zheng CY. L-Fuzzy unit interval and fuzzy connectedness. Fuzzy Sets Syst. 1988;27(1):73-76.