Can anyone please help me with this problem of point set topology?
Is the set A ={(x,y,z)| $x^{2} + y^2 - z^2=1$} compact in $R^3$?
I think its not compact and the way to go about it is to prove that the set isnt bounded. I tried setting some upper bound (a,b,c) for the terms but I have no clue how to proceed from there.
I'm already failing my course in real analysis. Any help will be much appreciated.
Solution 1:
You can rewrite the equation as $z^2 = x^2 + y^2 -1$. Then it should be clear that however large $x$ and $y$ are you can find $z \in \mathbb{R}$, which implies it is unbounded.