Normed Space of Fuzzy Intervals and Its Topological Structure

The space, Ƒcc(R), of all fuzzy intervals in R cannot form a vector space. However, the space Ƒcc(R) maintains a vector structure by treating the addition of fuzzy intervals as a vector addition and treating the scalar multiplication of fuzzy intervals as a scalar multiplication of vectors. The only difficulty in taking care of Ƒcc(R) is missing the additive inverse element. This means that each fuzzy interval that is subtracted from itself cannot be a zero element in Ƒcc(R). Although Ƒcc(R) cannot form a vector space, we still can endow a norm on the space Ƒcc(R) by following its vector structure. Under this setting, many different types of open sets can be proposed by using the different types of open balls. The purpose of this paper is to study the topologies generated by these different types of open sets.