Talk:Jordan curve theorem

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Illustration

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)[reply]

Question

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)[reply]

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)[reply]