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Is the following Idea well-known/usefull? The points of T not surrounded by A

  1. Dec 2, 2011 #1
    Is the following Idea well-known/usefull? The points of T not "surrounded" by A

    Let V be a metric space. Given a subset A[itex]\subseteq[/itex]V we define a new set A[itex]\widehat{}[/itex] to be equal to the set {x|for any open ball B1 with x [itex]\in[/itex]B1 there exists B2 s.t. B2 [itex]\subseteq[/itex] B1 and B2 [itex]\cap[/itex] A = ∅}.

    Examples: Let T be ℝ2. Let Δ = open unit ball. Then Δ[itex]\widehat{}[/itex] = ΔC (its compliment). Let Y = the line (t,t) t\inℝ then Y[itex]\widehat{}[/itex] = ℝ2.

    I'm thinking of A[itex]\widehat{}[/itex] as the points not "surrounded on all sides by A". The motivation for this definition is my attempt to solve the following problem: If f:ℝ→ℝ is smooth and given any x [itex]\in[/itex] ℝ fn(x) eventually vanishes for all n>k then f is a polynomial. If I could prove that the property BC = B[itex]\widehat{}[/itex] is preserved for some families of sets under chains of inclusions I can solve the above problem. However I'm interested in if this property is equivalent to something obvious I am missing.
     
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  3. Dec 2, 2011 #2

    Stephen Tashi

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    Re: Is the following Idea well-known/usefull? The points of T not "surrounded" by A

    Is B2 also an open ball?

    Do your course materials define an "interior point of A"? I think you have defined the complement of the interior of A.
     
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