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

1. Dec 2, 2011

### 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$\widehat{}$ = ΔC (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 BC = 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.

2. Dec 2, 2011

### Stephen Tashi

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.