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Which of the following subsets of $ \mathbb{R^2}$ are compact?

  1. $\displaystyle \{(x, y) : xy = 1\}$

  2. $\displaystyle \{(x, y) : x^{\large\frac{2}{3}} + y^{\large\frac{2}{3}} = 1\}$

  3. $\displaystyle \{(x, y) : x^2 + y^2 < 1\}$

    I am stuck on this problem. Can anyone help me please?

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    What is the definition of compact you are using? Do you have any theorems about compact sets which you think may be able to help you? If you include your thoughts on the problem, it is easier for people to write a response which will help you understand the answer.2012-12-27
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    Can you show that $[0, 1] \subset \mathbb{R}$ is compact using this definition?2012-12-27
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    If you know that a set in $\mathbb{R}^k$ is compact $\iff$ it is closed and bounded, then you should add this to your question. We have been assuming you need to use the definition you give in your comments. We have no way of knowing what you know unless you expand on your question a bit, so we can HELP you. As Michael originally asked: what theorems about compact sets can you use?2012-12-27
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    My thought process: "It is not a circle, but the equation has some superficial similarities. Circles are bounded. Maybe this is bounded." Is it?2012-12-27

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