Question

Let ( X , τ ) be a topological space, let A ⊆ X , and let x be an element of X. Show that the following two statements are equivalent. (a) x∈A−Bd(A)  (b) There is an open set U such that x∈U ⊆ A.

Answer 1

(a) has x as a member of an open set A because the boundary of A has been removed leaving the resultant set open. In (b) this open set has been defined as U so x must belong to it, by definition in (a) and U is a subset of the original set A, which is simply U with the boundary Bd(A) added.