What is the minimum epsilon radius for a closed set in an open set?

  • Thread starter ehrenfest
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In summary: But I don't understand what you're trying to do with this.In summary, the author is trying to find an epsilon radius for a closed set in an open set. He needs to find a way to take the infimal distance between the boundary of the closed set and the boundary of the open set, but he needs to prove that this is not zero.
  • #1
ehrenfest
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Homework Statement


I have a closed set in an open set in a metric space and I am trying to find an epsilon radius of the closed set that is in the open set. So I want to find some way to take the infimal distance between the boundary of the closed set and the boundary of the open set...but I need to prove that this is not zero...?

Homework Equations


The Attempt at a Solution

 
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  • #2
Why would the distance have to be infinitesimal? If I have a closed [itex]\epsilon[/itex]-ball, then wouldn't an open [itex]\frac{\epsilon}{2}[/itex]-ball do?
Now generalize to any closed set.
 
  • #3
Compuchip, I don't think "infimal" was meant to be "infinitesmal" but "infimum"- greatest lower bound of distances between the boundary of the closed set and boundary of the open set.
 
  • #4
Compuchip is right about what I am asking.
So, does anyone know how to do that?
 
  • #5
anyone?
 
  • #6
You need more than just closed, is it compact?
 
  • #7
Yes. It is closed an bounded. How do you use compactness here?
 
  • #8
The distance between a point in the closed set and the boundary of the open set is a nonzero, continuous function on a compact set. It attains a nonzero minimum.
 
  • #9
Maybe this will work. Get a nbhd of each of the points on the boundary of the closed set that is in the open set. Then take the interior of the closed set. So now we have an open cover of the closed set and we must have a finite subcover. Let x_i be the finite set of boundary points whose nbhds are in the finite subcover. Take the minimum of the radii of these nbhds. This will serve as the epsilon nbhd.

Does that work?

EDIT: I posted this before I read the post above. But still does this work?
 
  • #10
ehrenfest said:
Maybe this will work. Get a nbhd of each of the points on the boundary of the closed set that is in the open set. Then take the interior of the closed set. So now we have an open cover of the closed set and we must have a finite subcover. Let x_i be the finite set of boundary points whose nbhds are in the finite subcover. Take the minimum of the radii of these nbhds. This will serve as the epsilon nbhd.

Does that work?

EDIT: I posted this before I read the post above. But still does this work?

You are ok as far as the finite subcover, but then you lose me. There's not a finite set of bdy pts x_i, there is a finite set of open balls. It's easy to draw a picture where the bdy of the open set comes closer to an enclosed closed set than the minimum radius of a certain covering with balls.
 

1. What is a closed set within an open set?

A closed set within an open set is a subset of the open set that contains all of its boundary points. This means that the set includes all its limit points, or points that can be arbitrarily close to the set, as well as the points that are actually in the set.

2. How does a closed set relate to an open set?

A closed set is often described as the complement of an open set, meaning that it contains all the points that are not in the open set. Additionally, every open set can be represented as the complement of a closed set.

3. What is the significance of a closed set in an open set?

The concept of a closed set within an open set is important in topology, as it allows for the definition of continuity and convergence. It also helps to differentiate between different types of sets, such as open, closed, and compact sets.

4. Can a closed set be open?

No, a closed set cannot be open. By definition, a closed set contains all of its boundary points, while an open set does not. Therefore, a set cannot be both closed and open at the same time.

5. How is the concept of a closed set used in real-world applications?

The concept of a closed set within an open set is applicable in many areas of science and mathematics, such as in topology, analysis, and geometry. It is also used in practical applications, such as in engineering and computer science, for modeling and analyzing systems with continuous functions and limits.

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