Two types of bi-fuzzy open sets in bi-fuzzy topological spaces and their properties

Abstract

The idea of fuzzy topological spaces was initially proposed by Chang in 1968, inspired by Zadeh's discovery of fuzzy sets. In his work, the fundamental concepts of classical topology were examined to create a new theory of fuzzy topology integrating fuzzy sets. More recently, the notion of bi-fuzzy topological spaces has emerged. This study aimed to explore two novel types of open sets within bi-fuzzy topological spaces, namely, bi-fuzzy (𝑚,𝑛)-𝛼𝑛-open sets and bi-fuzzy (𝑚,𝑛)-𝛽𝑛-open, and elucidate some properties associated with these open sets. For a non-empty set 𝑋, a fuzzy topology is a family 𝜏 of fuzzy subsets in 𝑋 satisfying the following conditions: 0𝑋 ,1𝑋 ∈ 𝜏; finite intersection of members of 𝜏 is a member of 𝜏; and the arbitrary union of members of 𝜏 is a member of 𝜏. The pair (𝑋,𝜏) is called a fuzzy topological space. Also, the elements of 𝜏 are called fuzzy open sets. Moreover, we call the triple (𝑋,𝜏𝑚,𝜏𝑛) as the bi-fuzzy topological space, where 𝜏𝑚 and 𝜏𝑛 are two fuzzy topologies defined in a non-empty set 𝑋. Let (𝑋,𝜏𝑚,𝜏𝑛) be a bi fuzzy topological space, and 𝐴 be a fuzzy set. Then the set 𝐴 is called bi-fuzzy open if 𝐴 ∈ 𝜏𝑚 ∩𝜏𝑛; bi-fuzzy-(𝑚,𝑛)-preopen, if 𝐴 ≤ 𝑖𝑛𝑡𝑚(𝑐𝑙𝑛(𝐴)); bi-fuzzy-(𝑚,𝑛)-semi-open, if A≤𝑐𝑙𝑚(𝑖𝑛𝑡𝑛(𝐴)); bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open, if 𝐴 ≤ 𝑖𝑛𝑡𝑛(𝑐𝑙𝑚(𝑖𝑛𝑡𝑛(𝐴))); bi-fuzzy (𝑚,𝑛)-𝛽𝑛-open, if 𝐴 ≤ 𝑐𝑙𝑛(𝑖𝑛𝑡𝑚(𝑐𝑙𝑛(𝐴))). In this study, firstly it was demonstrated that a bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open set is distinct from a bi-fuzzy-(𝑚,𝑛)-𝛼𝑚-open set. Subsequently, we established that the union of two bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open sets is a bi fuzzy-(𝑚,𝑛)-𝛼𝑛-open set. However, the intersection of two bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open sets does not yield a bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open set. Furthermore, we illustrated that every bi fuzzy open set is a bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open. Also, every bi-fuzzy-(𝑚,𝑛)-𝛼𝑛-open set is bi-fuzzy-(𝑚,𝑛)-semi-open. However, the converse of these results is not true. Next, we showed that the union of two bi-fuzzy-(𝑚,𝑛)-𝛽𝑛-open sets is a bi-fuzzy-(𝑚,𝑛)-𝛽𝑛-open set. However, the intersection of two bi-fuzzy-(𝑚, 𝑛)-𝛽𝑛-open sets need not be a bi-fuzzy (𝑚,𝑛)-𝛽𝑛-open set. It is established that every bi-fuzzy open set is a bi-fuzzy-(𝑚, 𝑛)-𝛽𝑛 open. Finally, we showed that every bi-fuzzy-(𝑚,𝑛)-preopen set is bi-fuzzy-(𝑚,𝑛)-𝛽𝑛 open. In this study, within a bi-fuzzy topological framework, two new types of open sets were introduced, namely bi-fuzzy-(𝑚,𝑛)-𝛼𝑛open set and bi-fuzzy-(𝑚,𝑛)-𝛽𝑛-open sets. Also, the relationships between the union and intersection of these fuzzy sets were explained. Moreover, the connections between these two sets and the following categories were explored: bi-fuzzy-(𝑚,𝑛)-preopen sets, bi-fuzzy-(𝑚,𝑛)-semi-open sets and bi-fuzzy open sets.