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Proving a rectangle is connected.

  1. May 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Let K ={(x,y)[itex]\in[/itex]ℝ2:|x|≤1,|y|≤1}
    Prove that K is a connected subset of ℝ2

    3. The attempt at a solution

    Suppose f:[-2,2]→K and define f(x)={(x,y):|y|≤1}

    Dist(f(x),f(y))=sup(d(a,b):a[itex]\in[/itex]f(x),b[itex]\in[/itex]f(y))=d(x,y)=|x-y|. Using this equality it is easily shown that f(x) is continuous.

    So f is continuous and [-2,2] is connected therefore f([-2,2])=k is connected.

    Proving things are connected is very difficult and was just wondering if my proof was vaguely correct?
     
  2. jcsd
  3. May 5, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    What does that mean? What is f(0), for example?

    Here is a possible approach: for every two points (x1,y1), (x2,y2) in K, the straight line between them is part of K.
     
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