TITLE: Trouble understanding Borel sets definition
QUESTION [8 upvotes]: The definition in my book is:

The collection $B$ of Borel sets of real numbers is the smallest
  $\sigma$-algebra of sets of real numbers that contains all of the open
  sets of real numbers.

First, I'm a bit confused by the wording here.
Does this mean that if a collection $B$ is a $\sigma$-algebra and it contains all of the open sets of real numbers, then the sets in $B$ are called Borel sets?
I'm reading through some of the other answers to this question now, but it seems like different texts use different definitions, so if anyone could shed some light on the definition I provided, that would be helpful.

REPLY [2 votes]: First of all, for a Borel set, you need to have a topological space. Now by looking at the definition you have provided it seems that you are considering real line with standard topology. 
Explanation for your definition: A set $\beta $ is said to be a borel sigma algebra if the following two conditions are satisfied :

It contains all the open sets.
It is a sigma algebra and if $C$ is anny other sigma algebra containing all the open sets then $\beta \subset C$. (that is $\beta$ is smallest such set.) 

We will call elements of such a set to be borel sets.(observe that $\beta$ is a collection of sets)(for the intution you can think of it as if we are adding compliments and countable intersections of given open sets to the existing topology.)
In case of $\mathbb{R}$ with standard topology it is not that easy to find a set which is not a borel set. most of the sets which you can think of are all the borel sets. If you study measure theory you will eventually get a set which is not borel. 
If you are comfortable with topological spaces, and if you have got that borel sigma algebra depends on the topology, then i can give you a simple example of a non borel set in a different topological space just for understanding purpose. Pls comment to let me know.