Real Analysis/Interior, Closure, Boundary

Interior, Boundary, and Exterior

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

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

We denote ${\displaystyle ext(A)=\{x\in X:\exists \epsilon >0,B(x,\epsilon )\subset X\backslash A\}}$

Finally we denote ${\displaystyle br(A)=\{x\in X:\forall \epsilon >0,\exists y,z\in B(x,\epsilon ),{\text{ }}y\in A,z\in X\backslash A\}}$

Theorem

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

${\displaystyle int(A)\cup br(A)\cup ext(A)=X}$

${\displaystyle int(A)}$, ${\displaystyle br(A)}$, and ${\displaystyle ext(A)}$ are disjoint.

Closure

We denote ${\displaystyle cl(A)=A\cup Lim(A)}$

Theorem

${\displaystyle cl(A)=A\cup br(A)}$