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