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Relative openness

  1. Dec 7, 2009 #1
    1. The problem statement, all variables and given/known data
    T is open relative to X iff for any p [tex]\in[/tex] S there exists [tex]\delta[/tex] > 0 such that B(p,[tex]\delta[/tex] )[tex]\cap[/tex]X is [tex]\subset[/tex] T


    2. Relevant equations
    T is open relative to X provided there exists an open subset U of R^n such that T = U[tex]\cap[/tex]X


    3. The attempt at a solution
    Okay, so, going the forward ("if") direction, I think I'm able to classify this problem into one of three major subsets: X is closed, open, or neither. So, dealing with X being closed, I so far have:

    1. Pick p in T
    2. By hypothesis, produce U, an open subset of R^n s.t. T = U[tex]\cap[/tex]X
    3. For all points qn in U, let tau= min{|qn - p|}
    4.
    ...
    I believe that the final step will involve producing a delta ball within T, but I dn't know how to go about finding the correct delta. As well, if what I have so far is correct, then I think i'll have to end up letting delta be the minimum of tau and some other variable that I'll call beta for right now; so, how do I find the right beta?

    Does anyone have any ideas for the reverse ("only if") direction?

    Thanks guys!
     
    Last edited: Dec 7, 2009
  2. jcsd
  3. Dec 7, 2009 #2

    HallsofIvy

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    I don't see why you would need to think about whether X is open or closed or neither.

    If there exist an open subset U of R^n such that [itex]T = U\cap X[/itex], then, since U is open, there exist a ball [itex]B(p,\delta)\subset U[/itex] and so [itex]B(p, \delta)\cap X[/itex] is a subset of T. That's it.
     
  4. Dec 7, 2009 #3
    Conceptually, yes, I see what you are talking about. Can you give me another hint to push me along the path of proving it?

    Thanks for your help thus far
     
  5. Dec 7, 2009 #4
    Haha, well, during the five minutes after I wanted to ask you to give me a hint, I think i pretty much got it figured out....

    Forward direction: S is open relative to D if for any point p in S there exists delta >0 s.t. B(p,delta) intersect D is a subset of S

    1. Since S is open relative to D, produce U, a subset of R^n, s.t. S = U intersect D
    2. B/c U is open, for any point p in U, we can produce delta >0 s.t. B(p,delta) is a subset of U.
    3. Pick a point p in U intersect D
    4. Then, B(p,delta) is a subset of U intersect D, which is S

    Let me know if I forgot any details!
     
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