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Interior Points are Open

  1. Jan 16, 2013 #1
    1. The problem statement, all variables and given/known data
    For S [itex]\subset[/itex] Rn, prove that S° is open.

    2. Relevant equations
    S° are all interior points of S.

    3. The attempt at a solution
    My class has only learned how to use balls to solve these types of problems (no metric spaces). So I need to choose an ε > 0 so that Bε(x) [itex]\subset[/itex] S°, where x is any arbitrary point in S°. To show this is true, let y [itex]\subset[/itex] Bε(x) be arbitrary. (then I don't know how to progress further...how do I show that the neighbourhood contains only points in S°?)
     
  2. jcsd
  3. Jan 16, 2013 #2

    jbunniii

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    Start by choosing an arbitrary [itex]x \in S^o[/itex]. By definition this is an interior point of [itex]S[/itex], so there exists [itex]\epsilon > 0[/itex] such that [itex]B_\epsilon(x) \subset S[/itex]. Now, if you can show that every point [itex]y \in B_\epsilon(x)[/itex] is an interior point of [itex]S[/itex] then you're done. To do this, it certainly suffices to show that you can fit a smaller ball [itex]B_\delta(y)[/itex] around [itex]y[/itex] which is entirely contained within [itex]B_\epsilon(x)[/itex], because then you will have [itex]y \in B_\delta(y) \subset B_\epsilon(x) \subset S[/itex]. Try drawing a picture to see how to define [itex]\delta[/itex], the radius of the smaller ball.
     
  4. Jan 17, 2013 #3
    Thank you, I did not notice that you can put another ball inside the ball to make the proof work.

    Using this, I was able to make a series of inequalities using the triangle inequality, and managed to prove that [itex]y \in B_\delta(y) \subset B_\epsilon(x) \subset S[/itex].
     
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