there is a strengthening of the jordan curve theorem, called the jordan–schönflies theorem, which states that the interior and the exterior planar regions determined by a jordan curve in r are homeomorphic to the interior and exterior of the unit disk. in particular, for any point "p" in the interior region and a point "a" on the jordan curve, there exists a jordan arc connecting "p" with "a" and, with the exception of the endpoint "a", completely lying in the interior region. an alternative and equivalent formulation of the jordan–schönflies theorem asserts that any jordan curve "φ": "s" → r, where "s" is viewed as the unit circle in the plane, can be extended to a homeomorphism "ψ": r → r of the plane. unlike lebesgues' and brouwer's generalization of the jordan curve theorem, this statement becomes "false" in higher dimensions: while the exterior of the unit ball in r is simply connected, because it retracts onto the unit sphere, the alexander horned sphere is a subset of r homeomorphic to a sphere, but so twisted in space that the unbounded component of its complement in r is not simply connected, and hence not homeomorphic to the exterior of the unit ball.