Date: Jun 19, 2013 4:28 PM
Author: quasi
Subject: Re: Topology & Sigma Algebra

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