I have the following sets and I need to determine which are compact: $$ A = \{(x,y)| x^{2}+ y^{3} \leqslant 10\} $$ $$ B = \{(x,y)| x^{2}+ y^{4} \leqslant 10\} $$ $$ C = \{|x^{2} - 3y|: |x| \leqslant 1, 0\leqslant y \leqslant 1 \} $$
my thoughts: A is non bounded but closed so is non compact. B And C are both closed and bounded so they are compact sets.
Am I wrong? right? why?
You are right, $A$ is not bounded, and $B,C$ are both bounded and closed.
However, if that would be your answer, you would be lucky to get 20% of all possible points on an exam. You need to prove both that $A$ is unbounded (for example, by finding a sequence in $A$ that is unbounded, or a line) and that $B$ and $C$ are closed and bounded.