If you're not sure where to put it, report your own post and a mod will move it for you.
a). My guess is that your book defined "connected" as "simply connected", which is a bit odd. Either that, or it's a typo (or maybe a "thought-o"). I would have thought your answer correct, but I could definitely be wrong.
[EDIT]: See Plato and HallsofIvy's posts below for the correct explanation.
b). The real numbers are being considered as a subset of the complex plane, not just the real numbers like in real analysis. So, in looking at the neighborhoods around points in the real line, those neighborhoods are going to be two-dimensional. And you can see that any such neighborhood around a single point on the real line will have points in it that are not in the real line. So the real line is still closed, but it is no longer, as it is in real analysis, open. You see how that works? Essentially, the definition of "open set" has now changed, because your definition of "neighborhood" has changed.