1. The problem statement, all variables and given/known data Let X be a topological space. If A is a subset of X, the the boundary of A is closure(A) intersect closure(X-A). a. prove that interior(A) and boundary(A) are disjoint and that closure(A)=interior(A) union boundary(A) b. prove that U is open iff Boundary(U)=closure(U)-U c. If U is a nonempty open set in X, is it true that U=Interior(closure(U))? show it or give a counter example. 2. Relevant equations -the complement of a closed set is open - the closure of a set is defined as the intersection of all closed set containing A - the interior of a set is the union of all open sets contained in A 3. The attempt at a solution a. To prove that Int(A) and Boundary(A) are disjoint, assume the opposite. Let x be in both Int(A) and Bound(A). Then, if x is in Int(A), since Int(A) is open, there exists an open ngbd U of x such that U is contained in Int(A). Now, x in Bound(A) means that x is both in closure(A) and in closure(X-A). If x is in the closure of X-A, then every open ngbd of x intersects X-A. But, U is contained in A, so this is a contradiction. Thus Int(A) and Bound(A) are disjoint. (Am I missing anything here??) Now, I'm not sure how to prove this equality, although I understand it intuitively. I thought about doing it like this: Int(A) union Bound(A)=Int(A) union (clos(A) intersect clos(X-A))= (Int(A) union clos(A)) intersect (Int(A) unions clos(X-A))=clos(A) intersect (Int(A) union (clos(X-A)). Now, I just need to show that Int(A) union clos(X-A) =X, but I'm not sure how to do that. b. Assume Bound(U)=clos(U)-U, and U is not open. Then, U(complement) is not closed. Then, Bound(U)=clos(U) intersect U(complement) is not necessarily closed... it seems like this is getting a bit hazy.. Assume U is open. Then, clos(U)-U=clos(U)-Int(U)= clos(U) union (Int(U) complement)... again I need to show that Int(U) complement = clos(X-U) c. pretty sure this is true, don't quite know how to prove it Help please!