1. The problem statement, all variables and given/known data 2. Relevant equations A set is closed if it contains alll of its boundary points. A boundary point is a point where an open ball around that point has one point inside the ball that's in the set, and one point in the ball that's not in the set. 3. The attempt at a solution As seen in the picture, I can create infinitely many open balls around the edges of the triangle, with the balls containing at least one point in the triangle (red point) and one point outside the triangle (blue point). Since the perimeter of the triangle has all the boundary points, this triangle should be closed, but it's not. Why isn't it, if it contains all its boundary points?