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Topic: Topology & Sigma Algebra
Replies: 6   Last Post: Jun 21, 2013 11:52 AM

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 FredJeffries@gmail.com Posts: 1,676 Registered: 11/29/07
Re: Topology & Sigma Algebra
Posted: Jun 20, 2013 10:57 AM
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On Jun 20, 7:25 am, agapito6...@aol.com wrote:
> 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?

https://en.wikipedia.org/wiki/Borel_set#Non-Borel_sets

Date Subject Author
6/19/13 Agapito Martinez
6/19/13 quasi
6/20/13 Agapito Martinez
6/20/13 Agapito Martinez
6/20/13 FredJeffries@gmail.com
6/21/13 Agapito Martinez
6/21/13 FredJeffries@gmail.com

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