# Attempt to Define Region

• September 4th 2006, 07:32 PM
ThePerfectHacker
Attempt to Define Region
Can this be simplified? I am attempting to define what it means a region that has no holes in it because I never seen a definition of this. I am sure that differencial geometry offers one so if you know it please post. (I worked so hard on this eventhough it might seems basic).

Definition: Dualspace is $\mathbb{D}\subseteq \mathbb{R}^2$

Definition: The components of dualspace are the sets,
$S_x=\{x|(x,y)\in\mathbb{D}\} \mbox{ and }S_y=\{y|(x,y)\in \mathbb{D}\}$

Definition: Dualspace is bounded when its components have a lower and upper bound.

Definition: The component along $x_0\in S_x$ is the set, $S_x^{x_0}=\{y|(x_0,y)\in \mathbb{D}\}$. The component along $y_0\in S_y$ is the set, $S_y^{y_0}=\{x|(x,y_0)\in \mathbb{D}\}$.

Definition: The interval of existence along the components $S_x^{x_0} \mbox{ and }S_y^{y_0}$ are the closed intervals,
$[\min\{S_x^{x_0}\},\max\{S_x^{x_0}\}]$
$[\min\{S_y^{y_0}\},\max\{S_y^{y_0}\}]$ respectively.

Definition: A region is dualspace such that any element among the interval of existence is an element of the components of dualspace.
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My basic approach is that any vertical lines and horizontal lines are fully contained in a region. Although, the limititation of this definition is that it does include non-simple closed curves. However, it can be generalized to further dimensions.
• September 5th 2006, 04:23 PM
Plato
This response may be really not what you are looking for.
Topologist say that a region in $R^2$ has no holes if and only if each simple closed curve in the region can be ‘shrunk’ to a point. Alternatively, a bounded region has no holes if its boundary is a simple closed curve.
• September 5th 2006, 05:49 PM
ThePerfectHacker
Quote:

Originally Posted by Plato
This response may be really not what you are looking for.

I am sure it works, but it leaves questions unanswered. It was not what I was looking for. For example, then you need to define what a simple closed curve is.
• September 6th 2006, 04:51 PM
Plato
Quote:

Originally Posted by ThePerfectHacker
I am sure it works, but it leaves questions unanswered. It was not what I was looking for. For example, then you need to define what a simple closed curve is.

I must admit to being a bit surprised at that response.
The idea of a simple close curve is fundamental to the study of ‘holes’ in regions.
Have you read the popular piece in the Aug 28th New Yorker on the attempts to prove the Poincare’ problem? It is a great read!

A simple closed curve is simply the homemorphic image of the unit circle. So essentially a simple close curve can be ‘deformed’ into a unit circle! A good and readable source on all of this is Foundations of Modern Analysis by Dieudonne’. The basis for all of this is the Jordan Curve Theorem.
• September 6th 2006, 10:00 PM
CaptainBlack
Quote:

Originally Posted by Plato
I must admit to being a bit surprised at that response.
The idea of a simple close curve is fundamental to the study of ‘holes’ in regions.
Have you read the popular piece in the Aug 28th New Yorker on the attempts to prove the Poincare’ problem? It is a great read!

A simple closed curve is simply the homemorphic image of the unit circle. So essentially a simple close curve can be ‘deformed’ into a unit circle! A good and readable source on all of this is Foundations of Modern Analysis by Dieudonne’. The basis for all of this is the Jordan Curve Theorem.

I would recommend that you use bold for emphasis underlining makes
the emphasised text look too much like a link for my taste (I clicked on
your underlined text expecting to be taken to the article and/or the Amazon
page for the book)

RonL
• September 7th 2006, 04:04 AM
Plato
Quote:

Originally Posted by CaptainBlack
I would recommend that you use bold for emphasis underlining makes the emphasised text look too much like a link for my taste (I clicked on your underlined text expecting to be taken to the article and/or the Amazon page for the book) RonL

Sorry, but that was done out of habit. In North America it is customary to underline names of books and magazines.