On Wednesday, June 19, 2013 3:11:56 PM UTC-5, quasi wrote: > agapito6314 wrote: > > > > >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]. > > > > quasi
Thanks. Is B, in fact, the power set of R? If not, what elements of the power set would be missing from B?