The Jordan curve theorem was independently generalized to higher dimensions by
H. Lebesgue and
L. E. J. Brouwer in 1911, resulting in the
Jordan–Brouwer separation theorem. The proof uses
homology theory. It is first established that, more generally, if
X is homeomorphic to the
k-sphere, then the
reduced integral homology groups of
Y =
Rn+1 \
X are as follows: : \tilde{H}_{q}(Y)= \begin{cases}\mathbb{Z}, & q=n-k\text{ or }q=n, \\ \{0\}, & \text{otherwise}.\end{cases} This is proved by induction in
k using the
Mayer–Vietoris sequence. When
n =
k, the zeroth reduced homology of
Y has rank 1, which means that
Y has 2 connected components (which are, moreover,
path connected), and with a bit of extra work, one shows that their common boundary is
X. A further generalization was found by
J. W. Alexander, who established the
Alexander duality between the reduced homology of a
compact subset
X of
Rn+1 and the reduced cohomology of its complement. If
X is an
n-dimensional compact connected
submanifold of
Rn+1 (or
Sn+1) without boundary, its complement has 2 connected components. 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
R2 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:
S1 →
R2, where
S1 is viewed as the
unit circle in the plane, can be extended to a homeomorphism
ψ:
R2 →
R2 of the plane. Unlike Lebesgue's and Brouwer's generalization of the Jordan curve theorem, this statement becomes
false in higher dimensions: while the exterior of the unit ball in
R3 is
simply connected, because it
retracts onto the unit sphere, the
Alexander horned sphere is a subset of
R3 homeomorphic to a
sphere, but so twisted in space that the unbounded component of its complement in
R3 is not simply connected, and hence not homeomorphic to the exterior of the unit ball.
Arbitrary number of connected components Let K_0 and K_1 be homotopy equivalent compacts in \mathbb{R}^n. Then if \mathbb{R}^n\setminus K_i has a finite number of connected components for i=0,1, then so does \mathbb{R}^n\setminus K_{1-i}, and the two numbers coincide. If \mathbb{R}^n\setminus K_i has infinite number of connected components, so does \mathbb{R}^n\setminus K_{1-i}.
Discrete version The Jordan curve theorem can be proved from the
Brouwer fixed point theorem (in 2 dimensions), and the Brouwer fixed point theorem can be proved from the Hex theorem: "every
game of Hex has at least one winner", from which we obtain a logical implication: Hex theorem implies Brouwer fixed point theorem, which implies Jordan curve theorem. It is clear that Jordan curve theorem implies the "strong Hex theorem": "every game of Hex ends with exactly one winner, with no possibility of both sides losing or both sides winning", thus the Jordan curve theorem is equivalent to the strong Hex theorem, which is a purely
discrete theorem. The Brouwer fixed point theorem, by being sandwiched between the two equivalent theorems, is also equivalent to both. In reverse mathematics, and computer-formalized mathematics, the Jordan curve theorem is commonly proved by first converting it to an equivalent discrete version similar to the strong Hex theorem, then proving the discrete version.
Application to image processing In
image processing, a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of \Z^2. Topological invariants on \R^2, such as number of components, might fail to be well-defined for \Z^2 if \Z^2 does not have an appropriately defined
graph structure. There are two obvious graph structures on \Z^2: • the "4-neighbor square grid", where each vertex (x, y) is connected with (x+1, y), (x-1, y), (x, y+1), (x, y-1). • the "8-neighbor square grid", where each vertex (x, y) is connected with (x', y') iff |x-x'| \leq 1, |y-y'| \leq 1, and (x, y) \neq (x', y'). Both graph structures fail to satisfy the strong Hex theorem. The 4-neighbor square grid allows a no-winner situation, and the 8-neighbor square grid allows a two-winner situation. Consequently, connectedness properties in \R^2, such as the Jordan curve theorem, do not generalize to \Z^2 under either graph structure. If the "6-neighbor square grid" structure is imposed on \Z^2, then it is the hexagonal grid, and thus satisfies the strong Hex theorem, allowing the Jordan curve theorem to generalize. For this reason, when computing connected components in a binary image, the 6-neighbor square grid is generally used.
Steinhaus chessboard theorem The
Steinhaus chessboard theorem in some sense shows that the 4-neighbor grid and the 8-neighbor grid "together" implies the Jordan curve theorem, and the 6-neighbor grid is a precise interpolation between them. The theorem states that: suppose you put bombs on some squares on a n\times n chessboard, so that a king cannot move from the bottom side to the top side without stepping on a bomb, then a rook can move from the left side to the right side stepping only on bombs. == History and further proofs ==