## Proof That the Union of a Set and its Boundary Equals the Closure of the Set

Theorem

Ifis a set (open or closed) andis the boundary of the set then the closure oflabelled is equal to the union ofand the boundary oflabelledMore concisely,

Proof

SupposethenhenceSinceand

Suppose now thatIfthen good. Suppose then thatbut Each neighbourhood ofintersectsat a point distinct fromhencetherefore

Hence