(1)for $(x,y,z) \in \, \mathbb{R}^3, \quad x ^2 + y ^2 < 1, \quad z =0$
judge if this is open, closed, and bounded or not. and give proofs.
I'm not sure whether it's open or not. I think it's open, not closed and bounded. Because whatever neighbouhood you choose, it would certainly contain other points since z is always 0.then if you choose a open ball centered at any point, it would be 3 dimensional and go beyond the 2-d shape.
(2) judge whether a finite set is bounded, closed, open or not.Is finite set just some finite points?
I'm not sure about the answer either.
What do you guys think?