Now, I completely understand the definition of a Borel set (of real numbers) and I understand the definition of F_sigmas and G_deltas.

Now, directly after my book introduces these types of sets, a problem is given:

Given a lower semi-continuous function f defined for all reals, Royden asks, "What can you say about the sets {x: f(x)>a}, {x: f(x)>=a}, {x: f(x)<a}, {x: f(x)<=a}, and {x: f(x)=a}?"

I have already shown that the first set ( f(x)>a ) is open. Therefore, the set given by ( f(x)<=a ) is closed, since it's the compliment of the former. As for the other 3 sets, am I supposed to be gathering that they are F-sigmas or G-deltas or something? This question is vague, which I don't like.

I'm also having some trouble conceptualizing these Borel sets. Would anyone mind coming up with an example set of real numbers which is NOT a Borel set? Maybe that would, you know, help me get this. Thanks in advance for help on either of these two questions, and for any other words of wisdom you would deign to give.