Real Analysis/Interior, Closure, Boundary

Interior, Boundary, and Exterior

Let $A\subset X$, and $(X,d)$ a metric space.

We denote $int(A)=\{x\in X:\mathrm{\exists }ϵ>0,B(x,ϵ)\subset A\}$

We denote $ext(A)=\{x\in X:\mathrm{\exists }ϵ>0,B(x,ϵ)\subset X\mathrm{\setminus }A\}$

Finally we denote $br(A)=\{x\in X:\mathrm{\forall }ϵ>0,\mathrm{\exists }y,z\in B(x,ϵ),y\in A,z\in X\mathrm{\setminus }A\}$

Theorem

Let $A\subset X$, and $(X,d)$ be a metric space.

$int(A)\cup br(A)\cup ext(A)=X$

$int(A)$, $br(A)$, and $ext(A)$ are disjoint.

Closure

We denote $cl(A)=A\cup Lim(A)$

Theorem

$cl(A)=A\cup br(A)$