Restricted Jordan curve theorem

In summary, the conversation discusses the Restricted Jordan Curve theorem and the confusion regarding the relationship between nonintersecting segments being close and leaving a face by crossing C. The theorem states that a simple closed polygonal curve with a finite number of segments divides the plane into two faces, each with the curve as its boundary.
  • #1
ehrenfest
2,020
1

Homework Statement


http://ieeexplore.ieee.org/iel5/9518/30166/01385873.pdf?arnumber=1385873

I am reading a proof of the Restricted Jordan curve theorem (see Lemma 2 of the link) and the first two sentences are:

"Because the list of segments is finite, nonintersecting segments cannot be arbitrarily close. Hence we can leave a face only by crossing C."

I do not understand why the second sentence follows from the first.


Homework Equations





The Attempt at a Solution

 
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  • #2
I could not get to Lemma 2, only the abstract.
 
  • #3
What do you mean? It is on the top of page 6.
 
  • #4
You probably need a subscription to access that page (and such a subscription would likely be available through a university connection).
 
  • #5
Oh yeah. I forgot about the university subscription thing.

Restricted Jordan Curve Theorem: A simple closed polygonal curve C consisting of finite number of segments partitions the plane into exactly 2 faces, each have C as boundary.A polygonal curve is a curve composed of finitely many line segments.
 
  • #6
anyone?
 

FAQ: Restricted Jordan curve theorem

What is the Restricted Jordan curve theorem?

The Restricted Jordan curve theorem is a mathematical theorem that states that a simple closed curve in the plane divides it into two regions, an interior region and an exterior region. The curve cannot intersect itself, and it must be continuous.

Why is it called the "Restricted" Jordan curve theorem?

The "Restricted" in the name refers to the fact that the theorem only applies to simple closed curves, meaning curves that do not intersect themselves. The original Jordan curve theorem, which was proven by Camille Jordan in 1887, applies to any continuous closed curve in the plane.

What are some applications of the Restricted Jordan curve theorem?

The Restricted Jordan curve theorem has various applications in mathematics, physics, and engineering. It is used in topology to study the properties of continuous curves and surfaces. It also has applications in computer graphics and computer vision, where it is used to define and analyze shapes and contours. In physics, the theorem is used to study electromagnetic fields and their behavior around simple closed curves.

Can the Restricted Jordan curve theorem be extended to higher dimensions?

Yes, the theorem can be extended to higher dimensions. In fact, there are versions of the Jordan curve theorem for curves in higher-dimensional spaces, such as the three-dimensional space. However, the proof for higher dimensions is more complex and requires advanced mathematical concepts.

Are there any counterexamples to the Restricted Jordan curve theorem?

No, there are no known counterexamples to the Restricted Jordan curve theorem. However, there are variations of the theorem, such as the Jordan-Schönflies theorem, which apply to more general types of curves in the plane. These theorems have been proven to be equivalent to the Restricted Jordan curve theorem, meaning they are all true under the same conditions.

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