Real Analysis/Interior, Closure, Boundary

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Interior, Boundary, and Exterior[edit]

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[edit]

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[edit]

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

Theorem[edit]

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