Is this an alternative definition of a Borel set? I'm trying to get my head around Borel sets (like so many before me!) IF I've understood them correctly, the definition on http://en.wikipedia.org/wiki/Borel_set seems unnecessarily complicated. Isn't the following equivalent?
If for all x in A all values in the neighbourhood of x are also in A (in one direction, if x is the end of a closed interval) then A is a Borel set.
If this is not correct, what have I misunderstood?
 A: Sticking to the real line, the simplest definition of Borel sets is from the Borel $\sigma$-algebra which is defined to be the smallest $\sigma$-algebra which contains all open sets. To be clear, if $\sigma_\alpha$ is a sigma algebra containing the open sets, then you can take the intersection of all $\sigma_\alpha$ to form the Borel sigma-algebra (convince yourself that sigma-algebras are stable under intersection). 
If you want a constructive definition, start with the whole space and the empty set and then throw in every single open set and their complements. Then start taking unions and intersections and their complements. It sometimes doesn't pay to think about $\sigma$-algebras in this constructive way so sometimes it's more convenient to use the definition that they form the smallest sigma algebra. 
The definition you've stated seems to be for an open set, that it contains a neighborhood around every point. Be careful because open sets are NOT stable under intersection. In the comments the rationals were constructed with open sets but even more simply, think of $\{0\}=\bigcap\limits_{n=1}^\infty (-1/n,1/n)$. In other words $\{0\}$ is a Borel set even though it's not open. There are non-Borel sets out there (there's an example of one in your wikipedia link) but luckily they are pretty obscure.
