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Open set of real valued functions

  1. Feb 25, 2008 #1
    1. The problem statement, all variables and given/known data
    Consider the set X = {f:[0,1] [tex]\rightarrow[/tex] R | f [tex]\in[/tex] C[0,1]} w/ metric d(f,g) = sup|f(x) - g(x)| (x [tex]\in[/tex] [0,1])
    Prove that the set A = {f [tex]\in[/tex] X | f(0) > 1} is open in (X,d).



    2. Relevant equations
    E is open if every point of E is an interior point
    p is an interior point of E if there is a neighborhood N of p s.t. N[tex]\subset[/tex]E.
    C[0,1] is the set of all real valued functions on [0,1]
    supremum.

    3. The attempt at a solution

    To be honest, I have a great deal of difficulty understanding this question. Any hints or advice to help me in the right direction would be greatly appreciated.
     
  2. jcsd
  3. Feb 25, 2008 #2

    HallsofIvy

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    If f is a function in A what does a neigborhood of A look like?

    If A is the set of all functions, f, such that f(0)> 1, what are the interior points of A?
     
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