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If A ⊂ X, we define the boundary of A by the equation Bd A = A¯ ∩ (X - A). (a) Show that Int A and Bd A are disjoint, and A¯ = Int A ∪ Bd A. (b) Show that Bd A = ∅ ⇔ A is both open and closed. (c) Show that U is open ⇔ BdU = U¯ - U. (d) If U is open, is it true that U = Int(U¯) ? Justify your answer.

If A ⊂ X, we define the boundary of A by the equation Bd A = A¯ ∩ (X - A). (a) Show that Int A and Bd A are disjoint, and A¯ = Int A ∪ Bd A. (b) Show that Bd A = ∅ ⇔ A is both open and closed. (c) Show that U is open ⇔ BdU = U¯ - U. (d) If U is open, is it true that U = Int(U¯) ? Justify your answer.

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If A ⊂ X, we define the boundary of A by the equation Bd A = A¯ ∩ (X - A). (a) Show that Int A and Bd A are disjoint, and A¯ = Int A ∪ Bd A. (b) Show that Bd A = ∅ ⇔ A is both open and closed. (c) Show that U is open ⇔ BdU = U¯ - U. (d) If U is open, is it true that U = Int(U¯) ? Justify your answer.

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