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