Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

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


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
quasi

Posts: 10,195
Registered: 7/15/05
Re: Topology & Sigma Algebra
Posted: Jun 19, 2013 4:28 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.