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Math Help - Connected Sets in the Plane which are not Disjoint

  1. #1
    Member aldrincabrera's Avatar
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    Smile Connected Sets in the Plane which are not Disjoint

    ,.,hello there,.,.can anyone please help me with this???i'm so confused,.,.thnx

    Let A and B be Connected Sets in the Plane which are not Disjoint. is A intersection B necessarily connected?? What about A union B???
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  2. #2
    Super Member girdav's Avatar
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    Re: Connected Sets in the Plane which are not Disjoint

    Take A =\left\{(x,y)\in\mathbb R^2, 0\leq x\leq 1,0\leq y\leq 1\right\} and B=\left\{(x,x),0\leq x\leq 1\right\}. We have A\cap B =\left\{(0,0);(1,1)\right\}.
    For the second question, this lemma may help you: A topological space is connected if an only if every continuous map f:X\rightarrow \left\{0,1\right\} is constant.
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  3. #3
    Member aldrincabrera's Avatar
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    Re: Connected Sets in the Plane which are not Disjoint

    .,.thnx sir,.,ammm,.,.i've read something like this " a topological space X is said to be connected if there are no proper closed subsets A and B such that A intersection B = null and A union B = X..
    so i have somehow concluded that the first question is not necessarily connected. and the second is it is connected,.,.are my conclusions right sir??
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  4. #4
    Super Member girdav's Avatar
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    Re: Connected Sets in the Plane which are not Disjoint

    Your conclusions are right, now you have to show it.
    Did you try to prove the lemma?
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