
Re: Topology & Sigma Algebra
Posted:
Jun 21, 2013 9:58 AM


On Thursday, June 20, 2013 9:57:57 AM UTC5, FredJeffries wrote: > On Jun 20, 7:25 am, agapito6...@aol.com wrote: > > > On Wednesday, June 19, 2013 3:11:56 PM UTC5, 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? > > > > https://en.wikipedia.org/wiki/Borel_set#NonBorel_sets
Thanks for the link.

