(adsbygoogle = window.adsbygoogle || []).push({}); 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 B_{1}with x [itex]\in[/itex]B_{1}there exists B_{2}s.t. B_{2}[itex]\subseteq[/itex] B_{1}and B_{2}[itex]\cap[/itex] A = ∅}.

Examples: Let T be ℝ^{2}. Let Δ = open unit ball. Then Δ^{[itex]\widehat{}[/itex] }= Δ^{C}(its compliment). Let Y = the line (t,t) t\inℝ then Y^{[itex]\widehat{}[/itex]}= ℝ^{2}.

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 f^{n}(x) eventually vanishes for all n>k then f is a polynomial. If I could prove that the property B^{C}= 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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# Is the following Idea well-known/usefull? The points of T not surrounded by A

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