I have to check for compactness of given subets of $\mathbb{R}^2$.
$A =\{(x, y) :xy = 1\}$
$B =\{(x, y) :x^2y^2 = 1\}$
$C =\{(x, y) :e^x = \cos y\}$
$D =\{(x, y) :\mid x\mid +\mid y \mid \leq 10^{100}\}$
The purpose of asking above question is not just to get answers. I need concepts to deal with these kind of problems. Let me explain where I face difficulties.
Take set $A$; Intuitively this is clear to me that $A$ is not a compact subset of $\mathbb{R}^2$ as it is closed but unbounded. My problem is I am having trouble with checking boundedness or unboundedness of given subsets. Here I know that set $A$ consists of points which lies on rectangular hyperbola. So I have no difficulty in judging that set $A$ is unbounded. But I am not sure about others. Since I am not able to figure out set them.
Edit: I don't want graphical approach to solve these problems. Because quite often i face problems where I find myself unable to visualize graph of given functions. I think there must be available some mathematical tool to deal with this.
I need help to understand this. I would be very much thankful to all of you.