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$\subseteq$V we define a new set A^$\widehat{}$ to be equal to the set {x|for any open ball B₁ with x $\in$B₁ 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$\subseteq$V we define a new set A^$\widehat{}$ to be equal to the set {x|for any open ball B₁ with x $\in$B₁ 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.