The Euclidean topology on R (E) is that generated by the open intervals (x,y), closed under finite intersections and arbitrary unions. The Borel sigma algebra (B) also generated by the open intervals, is closed under complementation and countable intersections.
It appears as if some subsets of R are included in one and not the other. Is that the case? If so, can someone please supply examples of a set in E and not in B, and vice versa.