In this article, we will define the new notions (e.g.,
b
−
θ
-neighborhood system of point,
b
−
θ
-closure (interior) of a set, and
b
−
θ
-closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying
b
−
θ
-closed set is fuzzifying
γ
-closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying
b
−
θ
-derived set and fuzzifying
b
−
θ
-boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly
b
−
θ
-continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly
b
−
θ
-continuous mapping, fuzzifying strongly
b
−
θ
-irresolute mapping, and fuzzifying weakly
b
−
θ
-irresolute mapping along with different conditions for their existence are obtained.
A study on new type of ideal in topological spaces