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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.

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    As for boundedness, notice that in B and C, $x$ can be a very large negative number, while such a thing cannot happen in D, neither for $x$ nor for $y.$2012-06-06

2 Answers 2

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Draw pictures, or at least visualize. In set $B$, $x$ can be arbitrarily big. If you pick any $x\ne 0$, there is a $y$ such that $x^2y^2=1$. So there is no disk with centre the origin that contains all of $B$.

Also in set $C$, $x$ can be arbitrarily large negative. Just choose $y$ close to $\pi/2$, but a tiny bit below. Or else note that $y$ can be anything that makes $\cos y$ positive, and there are arbitrarily large $y$ with this property, such as $y=2n\pi$ where $n$ is any positive integer.

The set $D$ is clearly bounded, it is inside the disk with centre the origin and radius $10^{100}$. So you need to check whether or not $D$ is closed.

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    I did mistake. I forgot inequality given in problem. Now nothing to ask. Thank you very much. I learned new things from your comments and answers.2012-06-06
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For this you need the Heine-Borel Theorem which says that a subset of $\mathbb{R}^n$ is compact iff it is closed and bounded.

For $B$, note that the positive branch of $xy = 1$ is a subset of $B$. Since it is not bounded, $B$ fails to be compact.

For $D$ the constant $10^{100}$ is a red herring; the result is the same for any positive constant. Draw what $|x| + |y| \le 1$ and you will get the idea quickly.

For $C$, look at the graph of $x=\log(\cos(y))$

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    For C , can't we do without visualizing graph? That's what I want. Thanks for answering.2012-06-06