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Math Help - Show sets not connected

  1. #1
    Member thaopanda's Avatar
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    Show sets not connected

    Show that the following sets are not connected:
    (a) A := {(x,y) \in R^2 : x^2 + y^2 \neq 1};
    (b) B := {(x,y) \in R^2 : xy = 1}.

    For this problem, do I just need to show that A \cap B is the empty set?
    If so, I would need to prove that for any x,y \in A cannot equal any x,y \in B, right?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    Show that the following sets are not connected:
    (a) A := {(x,y) \in R^2 : x^2 + y^2 \neq 1};
    (b) B := {(x,y) \in R^2 : xy = 1}.

    For this problem, do I just need to show that A \cap B is the empty set?
    If so, I would need to prove that for any x,y \in A cannot equal any x,y \in B, right?
    You are not asked to show that A and B are disconnected from each other. (It seems like that's the way you interpreted the problem.) You are asked to show that A is disconnected. You are also asked to show that B is disconnected.

    A set X is disconnected if there exist two open sets, U and V such that:

    X\subset U\cup V and cl(U)\cap V=U\cap cl(V)=\emptyset

    --------------

    For (a), take U=\{(x,y):x^2+y^2<1\} and V=\{(x,y):x^2+y^2>1\}. It's obvious that A\subset U\cup V. It is also clear that cl(U)\cap V = U\cap cl(V)=\emptyset (simply because cl(U)=V^c and vice-versa).


    For (b), take U=\{(x,y):x>0,y>0\} and V=\{(x,y):x<0,y<0\}. Then proceed to show that:

    1. B\subset U\cup V
    2. cl(U)\cap V=U\cap cl(V)=\emptyset
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