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

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Agapito Martinez

Posts: 487
Registered: 12/13/04
Re: Topology & Sigma Algebra
Posted: Jun 20, 2013 10:21 AM
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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 for your help, this explains it.



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