Editing Jordan curve theorem (section)

=== 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.