Thumbnail: The blue circle represents the set of points (x, y) satisfying $$x^2 + y^2 = r^2$$. The red disk represents the set of points (x, y) satisfying $$x^2 + y^2 < r^2$$. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set. (Public Domain; Richard Giuly via Wikipedia)