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Proving that a subset is open

  1. Sep 7, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that the subset D={(x,y)| x≠0 and y≠0} is an open set in R^2
    .

    2. Relevant equations
    Open set: U is a subset of R^n. U is an open set when for every point X1, contained within U, there exists some open disk centered around X1 with radius r>0, that is completely contained within U. Or for simplicity's sake, a set U is open if it does not contain any of its boundary points.


    3. The attempt at a solution
    I have an understanding of what makes up an open set and know why this set is open, but i have no idea as to how I am meant to prove this, graphically or analytically( ideal method).
     
    Last edited: Sep 7, 2013
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  3. Sep 7, 2013 #2

    vela

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    What does the notation f(x;y) mean?
     
  4. Sep 7, 2013 #3
    Sorry about that, I miss quoted the question
     
  5. Sep 7, 2013 #4

    vela

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    First, tell us the definition of an open set.
     
  6. Sep 7, 2013 #5

    CompuChip

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    So let (x, y) be a point in D. Can you find a radius r such that all points in the open ball of radius r around this point are in D?

    First you may want to try drawing this, then proving it analytically. Note that for most points it's pretty trivial - the interesting case is for when (x, y) is close to (0, 0).
     
  7. Sep 7, 2013 #6

    Zondrina

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    Also, presuming you can always find ##r## for arbitrary ##(x,y) ≠ (0,0)##, what happens at ##(0,0)## exactly?

    Get reeeeeally close to ##0##. Can you always find an ##r## such that ##(0,0)## is not contained any neighborhood of your point?
     
    Last edited: Sep 7, 2013
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