The Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966), states that if
is a simple closed curve in , then has two components (an "inside" and "outside"),
with
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 (2002,
p. 169), or in classic texts such as Spanier (1966). Computer-checked proofs
of the theorem were constructed by a Japanese-Polish team in 2005 (Grabowski 2005)
and Hales (2007).
is a simple closed curve in 
, then 
has two components (an "inside" and "outside"),
with 
the boundary of each.
