near fixed point, near fixed interval circle and theirequivalence classes in a b−interval metric space

We introduce a novel distance structure called a b-interval metric space to generalize and extend metric interval space. also, we demonstrate that the collection of open balls, which forms a basis of a b-interval metric space, generates a t0-topology on it. further, we define topological notions like an open ball, closed ball, b-convergence, b-cauchy sequence and completeness of the space on a b-interval metric space to create an environment for the survival of a near fixed point and a unique equivalence class of near fixed point. towards the end, we introduce notions of interval circle, fixed interval circle, its equivalence class and established the existence of a near fixed interval circle and its equivalence interval c-class of near fixed interval circle to study the geometric properties of non-unique equivalence c-classes of nearly fixed interval circles.