Editing Jordan curve theorem (section)

{{Short description|A closed curve divides the plane into two regions}}
[[Image:Jordan curve theorem.png|thumb|Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "interior" region (light blue) and an "exterior" region (pink).]]

In [[topology]], the '''Jordan curve theorem''' ('''JCT'''), formulated by [[Camille Jordan]] in 1887, asserts that every ''[[Jordan curve]]'' (a plane simple closed curve) divides the plane into an "interior" region [[Boundary (topology)|bounded]] by the curve (not to be confused with the [[interior (topology)|interior]] of a set) and an "exterior" region containing all of the nearby and far away exterior points. Every [[path (topology)|continuous path]] connecting a point of one region to a point of the other intersects with the curve somewhere. 

While the [[theorem]] seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." ({{harvtxt|Tverberg|1980|loc=Introduction}}). More transparent proofs rely on the mathematical machinery of [[algebraic topology]], and these lead to generalizations to [[higher-dimensional space]]s.

The Jordan curve theorem is named after the [[mathematician]] [[Camille Jordan]] (1838–1922), who published its first claimed proof in 1887.{{sfnp|Jordan|1887}}<ref>{{cite journal
 | last = Kline | first = J. R.
 | doi = 10.2307/2303093
 | journal = American Mathematical Monthly
 | mr = 6516
 | pages = 281–286
 | title = What is the Jordan curve theorem?
 | volume = 49
 | year = 1942| issue = 5
 | jstor = 2303093
 }}</ref> For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by [[Oswald Veblen]]. However, this notion has been overturned by [[Thomas Callister Hales|Thomas C. Hales]] and others.<ref>{{cite journal
 | last = Hales | first = Thomas C. | author-link = Thomas Callister Hales
 | department = From Insight to Proof: Festschrift in Honour of Andrzej Trybulec
 | issue = 23
 | journal = Studies in Logic, Grammar and Rhetoric
 | publisher = University of Białystok
 | title = Jordan's proof of the Jordan curve theorem
 | url = https://www.maths.ed.ac.uk/~v1ranick/papers/hales1.pdf
 | volume = 10
 | year = 2007}}</ref>