Editing Jordan curve theorem (section)

== Proof and generalizations ==

The Jordan curve theorem was independently generalized to higher dimensions by [[H. Lebesgue]] and [[L.&nbsp;E.&nbsp;J. Brouwer]] in 1911, resulting in the '''Jordan–Brouwer separation theorem'''.

{{math theorem|math_statement=
Let ''X'' be an ''n''-dimensional ''[[topological sphere]]'' in the (''n''+1)-dimensional [[Euclidean space]] '''R'''<sup>''n''+1</sup> (''n'' > 0), i.e. the image of an injective continuous mapping of the [[n-sphere|''n''-sphere]] ''S<sup>n</sup>'' into '''R'''<sup>''n''+1</sup>.  Then the complement ''Y'' of ''X'' in '''R'''<sup>''n''+1</sup> consists of exactly two connected components.  One of these components is bounded (the interior) and the other is unbounded (the exterior).  The set ''X'' is their common boundary.
}}

The proof uses [[homology theory]]. It is first established that, more generally, if ''X'' is homeomorphic to the ''k''-sphere, then the [[reduced homology|reduced integral homology]] groups of ''Y'' = '''R'''<sup>''n''+1</sup> \ ''X'' are as follows:

: <math display="block">\tilde{H}_{q}(Y)= \begin{cases}\mathbb{Z}, & q=n-k\text{ or }q=n, \\ \{0\}, & \text{otherwise}.\end{cases}</math>

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 [[James Waddell Alexander II|J. W. Alexander]], who established the [[Alexander duality]] between the reduced homology of a [[compact space|compact]] subset ''X'' of '''R'''<sup>''n''+1</sup> and the reduced cohomology of its complement. If ''X'' is an ''n''-dimensional compact connected submanifold of '''R'''<sup>''n''+1</sup> (or '''S'''<sup>''n''+1</sup>) 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 '''R'''<sup>2</sup> 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 ''&phi;'': ''S''<sup>1</sup> → '''R'''<sup>2</sup>, where ''S''<sup>1</sup> is viewed as the [[unit circle]] in the plane, can be extended to a homeomorphism ''&psi;'': '''R'''<sup>2</sup> → '''R'''<sup>2</sup> 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 '''R'''<sup>3</sup> is [[simply connected]], because it [[deformation retract|retracts]] onto the unit sphere, the [[Alexander horned sphere]] is a subset of '''R'''<sup>3</sup> homeomorphic to a [[sphere]], but so twisted in space that the unbounded component of its complement in '''R'''<sup>3</sup> is not simply connected, and hence not homeomorphic to the exterior of the unit ball.

=== Discrete version ===
The Jordan curve theorem can be proved from the [[Brouwer fixed-point theorem|Brouwer fixed point theorem]] (in 2 dimensions),{{sfnp|Maehara|1984|p=641}} and the Brouwer fixed point theorem can be proved from the Hex theorem: "every [[Hex (board game)|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.<ref>{{Cite journal |last=Gale |first=David |date=December 1979 |title=The Game of Hex and the Brouwer Fixed-Point Theorem |journal=The American Mathematical Monthly |volume=86 |issue=10 |pages=818–827 |doi=10.2307/2320146 |jstor=2320146 |issn=0002-9890}}</ref>

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 mathematics|discrete]] theorem.

The Brouwer fixed point theorem, by being sandwiched between the two equivalent theorems, is also equivalent to both.<ref>{{Cite book |last1=Nguyen |first1=Phuong |last2=Cook |first2=Stephen A. |title=22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007) |chapter=The Complexity of Proving the Discrete Jordan Curve Theorem |date=2007 |pages=245–256 |publisher=IEEE |doi=10.1109/lics.2007.48|arxiv=1002.2954 |isbn=978-0-7695-2908-0 }}</ref>

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.<ref>{{Cite journal |last=Hales |first=Thomas C. |date=December 2007 |title=The Jordan Curve Theorem, Formally and Informally  |journal=The American Mathematical Monthly |volume=114 |issue=10 |pages=882–894 |doi=10.1080/00029890.2007.11920481 |s2cid=887392 |issn=0002-9890}}</ref>

==== Application to image processing ====
In [[Digital image processing|image processing]], a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of <math>\Z^2</math>. Topological invariants on <math>\R^2</math>, such as number of components, might fail to be well-defined for <math>\Z^2</math> if <math>\Z^2</math> does not have an appropriately defined [[Pixel connectivity#Types of connectivity|graph structure]].

There are two obvious graph structures on <math>\Z^2</math>: 
[[File:Sasiedztwa_4_8.svg|right|thumb|8-neighbor and 4-neighbor square grids.]]
* the "4-neighbor square grid", where each vertex <math>(x, y)</math> is connected with <math>(x+1, y), (x-1, y), (x, y+1), (x, y-1)</math>.
* the "8-neighbor square grid", where each vertex <math>(x, y)</math> is connected with <math>(x', y')</math> iff <math>|x-x'| \leq 1, |y-y'| \leq 1</math>, and <math>(x, y) \neq (x', y')</math>.

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 <math>\R^2</math>, such as the Jordan curve theorem, do not generalize to <math>\Z^2</math> under either graph structure.

If the "6-neighbor square grid" structure is imposed on <math>\Z^2</math>, 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.<ref>{{Cite web |last=Nayar |first=Shree |date=Mar 1, 2021 |title=First Principles of Computer Vision: Segmenting Binary Images {{!}} Binary Images |website=[[YouTube]] |url=https://www.youtube.com/watch?v=2ckNxEwF5YU&ab_channel=FirstPrinciplesofComputerVision}}</ref>

==== 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.<ref>{{Cite journal |last=Šlapal |first=J |date=April 2004 |title=A digital analogue of the Jordan curve theorem |journal=Discrete Applied Mathematics |volume=139 |issue=1–3 |pages=231–251 |doi=10.1016/j.dam.2002.11.003 |issn=0166-218X|doi-access=free }}</ref><ref>{{cite journal
 | last = Surówka | first = Wojciech
 | issue = 7
 | journal = Annales Mathematicae Silesianae
 | mr = 1271184
 | pages = 57–61
 | title = A discrete form of Jordan curve theorem
 | url = https://rebus.us.edu.pl/handle/20.500.12128/14250
 | year = 1993}}</ref>

The theorem states that: suppose you put bombs on some squares on a <math>n\times n</math> 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.