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Ball is a Jordan Region

  1. Apr 12, 2013 #1
    I was trying to show that any ball in ℝn is a Jordan region, and this amounts to showing that its boundary has volume zero (Jordan content 0).

    My TA proposed that you break the circle up into pieces of lines, and then adapt the following proof.

    [
    Pf: A Horizontal line of length 1 has content 0

    A horizontal line of length 1 in ℝ2 has volume 0 because if you cover it by n intervals of length ε, you see that nε=1 (or something like 1+ε*γ for some γ<1, if ε doesn't divide 1).

    So, then define a grid of squares with sides ε. Then the volume of the grid, which bounds the volume of the circle, is ε2*n but since nε=1 (or about 1), you get ε2*(1/ε)=ε, and so we can bound the volume by an arbitrary ε, so the boundary of the ball has content 0.
    ]

    The TA actually did this with balls instead of squares (i.e. balls of radius ε), but since we're trying to prove that a ball is a Jordan region, I wasn't sure about this.


    Anyway, I was not sure how to adapt this proof to the curved line, since there is no guarantee that the curved line will only pass through n of the squares of volume ε2. I've seen at least 1 other way to do this with the compactness of the circle, but I was hoping that I could understand the TA's way.

    Thanks.
     
    Last edited: Apr 12, 2013
  2. jcsd
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