Talk:Jordan curve theorem

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 Field: Topology

Formal proof[edit]

I have just clarified the formal proof priority, sorry for doing it anonymously - have not noted I was logged out. JosefUrban (talk) 15:41, 4 January 2009 (UTC)

Illustration[edit]

An illustration could really improve the article and give credence to the claim that the concept is intuitive.--Cronholm144 10:14, 25 May 2007 (UTC)

This has been done.
98.67.108.12 (talk) 16:23, 25 August 2012 (UTC)

Question[edit]

Hello, I have a question. A book that I am reading says that "A Jordan curve is an equivalence class of homeomorphisms of I into R2 (or of S1 into R2 in the case of closed curves)." This article defines a Jordan curve as "a simply closed curve."

1. The book I am reading is implying an arbitrary equivalence relation? And what is the purpose of this equivalence class? (I assume that it is just a way of saying 'all the identical homeomorphisms', so that it gets all the same shapes that are represented differently?)

2. This article says that a Jordan curve is a simple closed curve, but I think the definition my book gave says that it doesn't have to be simply closed, though it may, i.e. 'a homeomorphism of S1 in the case of closed curves'. But I know that this article is right, because a Jordan curve is defined identically in Curve.

Sorry for the stupid questions. Great article!

Veritas (talk) 16:37, 22 September 2008 (UTC)
As for 1, the book certainly refers to some specific equivalence relation. You have to look backwards for a definition of equivalent curves. It is hard to guess what they mean by it without reading it, though one possibility is that curves are equivalent if they differ by a homeomorphic change of parametrization.
As for 2, note first that a homeomorphism is in this context the same thing as an injective continuous map (because S1 is compact, and R2 is Hausdorff), thus the two definitions agree on what closed curves are Jordan curves. As far as I am aware, allowing Jordan curves to be non-closed is highly unusual. At any rate, the Jordan curve theorem only applies to closed curves. — Emil J. 15:33, 23 September 2008 (UTC)
To the original questioner: You do need to allow for the same thing having more than one name. Sometimes, there is even a very technical name and a common name. For those of us who have studied complex analysis and/or real calculus of two dimensions, "simple closed curve" tells it all. We know what those are.
In some kinds of mathematics on the Ph.D. level, some more precise wording might be necessary. Do not get "wrapped around the axle" about it.
98.67.108.12 (talk) 16:36, 25 August 2012 (UTC)

Applications in collision detection[edit]

Hi guys.... I think this article needs to be expanded to encompass one of its most useful applications: 2D collision detection. These sites give a good explanation of what the Jordan Curve Theorem means in a practical sense, and would help explain the concept to the less math-inclined. The strategy: http://tog.acm.org/editors/erich/ptinpoly/ Example implementation: http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html —Preceding unsigned comment added by Oticon6 (talkcontribs) 14:14, 1 April 2009 (UTC)

suggested addition[edit]

This paper seems like a nice reference. It has nice pictures in it. I hope one of you experts will include it somehow :)

https://facultystaff.richmond.edu/~wross/PDF/Jordan-revised.pdf — Preceding unsigned comment added by 98.249.0.238 (talk) 00:37, 31 March 2012 (UTC)

OK, I did. :-) Boris Tsirelson (talk) 17:44, 31 March 2012 (UTC)

So, what about "M. Reeken"?[edit]

Now, he or she was "M. Reeken", and I added the "M", which is given in the references. So, "Who Reeken"? Please give the whole name.
Just calling someone "Reeken" or "Jordan" or "Jones" or "Schwartz" on first appearace is actually quite rude. If this is a person who wants to be referred to as "M. Reeken", then please give her or him that much. As for the present, we do not know if this was "Marion Reeken", "Michael Reeken", "Michelle Reeken", or "Madison Reeken".
98.67.108.12 (talk) 16:29, 25 August 2012 (UTC)

Michael, per the reference. Not that I know anything about him.—Emil J. 12:48, 27 August 2012 (UTC)

Contribution of A.S. Zoch - is it notable?[edit]

An anon "IP 67.226.22.24" just added two phrases:

  • The difficult part of the proof of the Jordan Curve theorem lies in the fact that the two disjoint components of the plane cut by the given curve are open and the curve itself is closed.
  • The Jordan Polygonal theorem was presented at ICM 1998 by A. S. Zoch and soon afterwards a simple proof of the Jordan Curve theorem was presented to AMS by the same author using linear lemmas.

About the latter phrase. As far as I know, this work by Zoch was presented on several conferences (including ICM 1998 and ICM 2002) in the form of short communication or contributed paper, but was never published in mathematical literature. Was it refereed? Is it notable?

About the former phrase. Does it make sense?

Boris Tsirelson (talk) 20:06, 28 February 2013 (UTC)

I’d say the answers are no and no.—Emil J. 13:07, 1 March 2013 (UTC)

So you were not there otherwise you may know the difference. Perhaps A. Zoch would explain things to you if you cared more about mathematics. A. S. Zoch has publications in the areas of mathematics, colour theory, optics, encryption, and light. — Preceding unsigned comment added by 67.226.62.182 (talk) 19:16, 9 June 2016 (UTC)

I have question[edit]

In a course on Complex Analysis I attended this Jordan Curve theorem was referenced with using curve Γ, which divides the plane to two regions which are bordered by Γ and which one is interior region(bounded) and another unbounded region is exterior region( but these regions have NO formal notation like I show following ) And I know also that at least I have used for set A, interior region as I(A) and exterior region E(A). Is this right use? I mean is it wrong that you would not define interior region by the edge of A as closed jordan curve C? So Interior region would be I(A). And is it wrong to not to define exterior region by this C? So exterior region would be E(A). — Preceding unsigned comment added by Alvoutila (talkcontribs) 12:58, 15 August 2014 (UTC)

My question too! Phrased more simply: does the curve belong to the interior (a closed or compact set), the exterior (open set), or neither? The notion of crossing the boundary implies that both interior and exterior sets are open, and only the boundary is compact.220.240.251.114 (talk) 01:09, 4 May 2016 (UTC)

Sorry, I do not understand this question. The Jordan curve is compact. Its complement is open. This complement is the union of two disjoint connected open sets, one bounded ("the interior") and one unbounded ("the exterior").
Ah, well, maybe I do understand. This use of the word "interior" is not its use in the article "interior (topology)". Somewhat misleading terminology, indeed. Boris Tsirelson (talk) 17:55, 4 May 2016 (UTC)

Translation?[edit]

Does the cited 4-page proof of Filippov have an English translation anywhere?? (If not, could someone make students translate that proof on their math gradschool language exam and post the results online??) It is a horrible tease to be directed to an "elementary" proof, only to find that it is in Russian. — Preceding unsigned comment added by 173.66.241.122 (talk) 19:07, 8 May 2016 (UTC)