Editing Jordan curve theorem (section)

==== 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>