Ok, I am needing a little help with open and closed sets.

{1, 1/2, 1/3, 1/4, 1/5,....}

I am pretty sure that this is NOT closed since this accumulates to 0 but 0 is not included in the set. I also think that this is NOT open since 1 is a boundary point and an open set cannot contain a boundary point. Thus I think this one is neither open nor closed, am I correct?

(0,1)U(1,2)U(2,3)U(3,4)U...U(n, n+1)U...

This one has me confused. It seems like it would be open since it contains all open intervals, but then again it seems like it has no interior points since it just seems to go on forever the way the natural numbers do. Thus, I really do not know if it is open or not. However, I do feel like it is closed since it does not appear to have an accumulation point. Help?

{x: x²<2}

Is this the same as [-1,1]? If so then I think this is closed because every closed interval is a closed set. I also think that it is not open.

Please help me out, I would like to know what I've gotten right and what I've gotten wrong. Thanks!!