I am working on following mathematical problem:
Check for compactness of following subets of $\mathbb{R}^2$.
$A =\{(x, y) :xy = 1\}$
$B =\{(x, y) :x^2y^2 = 1\}$
My attempt: I know that all I have to check whether subsets $A$ and $B$ are closed and bounded in $\mathbb{R}^2$ or not?
Checking for boundedness: Can we say that subsets A and B are not bounded in $\mathbb{R}^2$ since they contains points $(n, 1/n)$, where $x = n \in \mathbb{N}$ tends to infinity. I am not entirely confident about the correctness of my approach. Could you please review my approach and let me know if I am on the right track or not? Additionally, if you have any alternative approaches to suggest, I would greatly appreciate your insights.
Checking for closedness
I feel both sets are closed but how can I proved that?
PS: I have asked the same question before, but this time I have some additional doubts regarding my approach. Thank you very much
Yes, both sets are closed and both sets are not bounded, hence they are not compact.
For the boundedness, your idea works. $(n,1/n)$ belongs to $A$ for all $n$, hence the norm of the elements of $A$ is not bounded. Similarly for $B$ with the same counterexample.
For closedness, just notice that $A=f^{-1}(\{1\})$ where $f:\mathbb R^2\to \mathbb R$ is the map $f(x,y):=xy$, which is continuous. Again, similarly for $B$ with the map $g(x,y):=x^2y^2$.