Recent content by c16

Homework Statement I need to prove that the int(U union Bdy(U))=Int(U) when U is open. Homework Equations Bdy(U)=closure(U) intersect closure(X-U) a point is in the interior if there is an open neighborhood of the point that is contained in the set. The Attempt at a Solution...

I think I've got it all, except for the last part of part c. I'm trying to show that if S=U union bdy(U), then Int(S)=U. Remember that U is open. Obviously, I have no trouble showing that if x is in U, x is in Int(S). However, how do I show that if x is in Bdy(U), x is not in Int(S)? I've shown...

thanks so much for your help with c! I'd like to try to prove that int is distributive across unions eventually, but would you help me with part (a) first? I have that Int(A) and Bdy(A) are disjoint, and now I need to prove that Closure(A)=Int(A) union Bdy(A). I know that if x is in Int(A), x is...

I'm not sure how C) helps me... so if closure(U)=Interior(U) union Bound(U), then Int(closure(U))=Int(Int(U) union Bound(U))=Int(U union Bound(U)) because U is open. Not sure where to go from here... what I really need to show is that Int(U)=Int(Closure(U)).

Homework Statement 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...