Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Agapito Martinez Posts: 487 Registered: 12/13/04
Re: Topology & Sigma Algebra
Posted: Jun 21, 2013 9:58 AM

On Thursday, June 20, 2013 9:57:57 AM UTC-5, FredJeffries wrote:
> 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