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Math Help - how to prove a closed set in Rⁿ

  1. #1
    Junior Member sorv1986's Avatar
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    Thumbs up how to prove a closed set in Rⁿ

    hi!

    what is the easiest way to prove that the set A is closed and the set B is not closed?

    1). A={(x,y) ∈ R: x+y=1}
    and
    2). B={(x,y) : ∈ R: x - y=1}



    Help please. thanks please
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sorv1986 View Post
    hi!

    what is the easiest way to prove that the set A is closed and the set B is not closed?

    1). A={(x,y) ∈ R: x+y=1}
    and
    2). B={(x,y) : ∈ R: x - y=1}



    Help please. thanks please
    I mean, the easiest way would be to note that f:\mathbb{R}^2\to\mathbb{R}, f(x,y)=x^2+y^2 is continuous (it's just a polynomial) and A=f^{-1}({1}) and for the second one are you sure it isn't closed?
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  3. #3
    Junior Member sorv1986's Avatar
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    kolkata, india
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    thanks for ur response sir

    oho....actually the 2nd one i think its a hyperbola...so it should be not closed...am i wrong?.....but in the first case...i did not understand what r u actually doing to prove that set A is closed
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