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Math Help - bounded,open,closed or connected?

  1. #1
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    bounded,open,closed or connected?

    Hey. This was a question I came across. What is the right answer?

    If A and B are subsets of the Euclidean space,
    and A + B ={x + y | x Є A and y Є B}. Then which of the following statements is false?

    a) If A and B are bounded, then A + B is bounded
    b) If A and B are closed, then A + B is closed
    c) If A and B are open, then A + B is open
    d) If A and B are connected, then A + B is connected.

    I think options (a) and (b) are right. I am not so sure about openness and connectedness..
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  2. #2
    Senior Member Shanks's Avatar
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    all are right except (b).
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  3. #3
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    Quote Originally Posted by Shanks View Post
    all are right except (b).
    Oh can you tell me how b is wrong?
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  4. #4
    Senior Member Shanks's Avatar
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    A counterexample: Let N be the set of all positive integers.
    A=N, B=\left\{\frac{1}{n}-n: n\in N, 2\le n\right\}
    A, and B are closed, but A+B has 0 as a cluster point, and 0 is not contained in A+B.
    Last edited by Plato; December 28th 2009 at 06:51 AM. Reason: LaTeX
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