Real Analysis/Interior, Closure, Boundary

Interior, Boundary, and Exterior

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

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

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

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

Theorem

Let $A \subset X$, 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

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