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

[deluks917]

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\subseteqV we define a new set A^\widehat{} to be equal to the set {x|for any open ball B₁ with x \inB₁ there exists B₂ s.t. B₂ \subseteq B₁ and B₂ \cap A = ∅}.

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

I'm thinking of A^\widehat{} 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 \in ℝ fⁿ(x) eventually vanishes for all n>k then f is a polynomial. If I could prove that the property B^C = B^\widehat{} 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.

 

[Stephen Tashi]

Let V be a metric space. Given a subset A\subseteqV we define a new set A^\widehat{} to be equal to the set {x|for any open ball B₁ with x \inB₁ there exists B₂ s.t. B₂ \subseteq B₁ and B₂ \cap 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.