If J is a simple closed curve in R^2, then the Jordan curve theorem, also called the Jordan-Brouwer theorem states that R^2 - J has two components (an "inside" and "outside"), with J the boundary of each. The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher, or in classic texts such as Spanier. Computer-checked proofs of the theorem were constructed by a Japanese-Polish team in 2005 and Hales.

Jordan curve | Schönflies theorem