Complex analysis - connected sets

*Note:* Sorry if this is in the wrong section, I wasn't sure where to put it.

**The question**

Are the following regions in the plane 1) open 2) connected, 3) domains?

a) the complement of the unit circle

b) the real numbers

**My attempt**

a) yes, yes, no

However the solution is yes, no, no. I don't understand why this isn't considered connected. The way I see it, the compliment of the unit circle is everything outside the circle. So there's always a 'path' from one element in the set to another, if I'm not mistaken. Am I missing a detail here?

b) yes, yes, yes

The correct solution is no, yes, no. Why are the real numbers considered a closed set? I'm a little confused, since real numbers can be 'as big as you like'. I think I'm missing some intuition here. I'm also not sure why it isn't considered a domain, but I guess that's because I believe it's open. I realise it can't be a domain if it's not both open and connected.

Any assistance would be greatly appreciated!