Proving the Interior of a Boundary for Open Sets

[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

obviously, if x is in U and U is open, there is a neighborhood of x in U by the fact that U is open. If x is in Bdy(U), then I want to prove that every open neighborhood of x is not in S=U union Bdy(U). Now, by the definition of boundary, every open neighborhood of x intersects X-U, so obviously there are points outside of U. What I need to prove is that there is a point y in every open neighborhood of x that is not in U AND is not in the closure of U, meaning that it is not in the boundary.

 

[Dick]

Take U to be the union of the open intervals (0,1) and (1,2). What you are trying to prove is false.