ABSTRACT
A curve is the image of a continuous function from [0, 1] to the plane (or surface). The curve is simple if the function is injective, and it is closed if its two endpoints are the same. We use the term simple closed curve to denote a closed curve which is simple except for its two endpoints, which are the same. Equivalently, a simple closed curve is a curve which is homeomorphic to the unit circle. While curves in general can behave rather awkwardly (they can be space-filling, for example), simple curves are more well-behaved as witnessed by the following two theorems. A subset of the plane or surface is (arc-)connected if any two points in the set can be connected by a curve belonging to the set. A maximal connected subset is called a region.
