>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.
First, note that E doesn't need arbitrary unions -- countable unions suffice (every open interval contains a rational number).
Also B gets countable unions via DeMorgan's law, hence B contains all open sets.
Thus, E is a subset of B.
However B contains sets which not open, hence not in E, for example [0,1].