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!