• Support PF! Buy your school textbooks, materials and every day products Here!

Closed set in an open set

  • Thread starter ehrenfest
  • Start date
  • #1
2,012
1

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

 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
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
HallsofIvy
Science Advisor
Homework Helper
41,770
911
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
2,012
1
Compuchip is right about what I am asking.
So, does anyone know how to do that?
 
  • #5
2,012
1
anyone?
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618
You need more than just closed, is it compact?
 
  • #7
2,012
1
Yes. It is closed an bounded. How do you use compactness here?
 
  • #8
Dick
Science Advisor
Homework Helper
26,258
618
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
2,012
1
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
Dick
Science Advisor
Homework Helper
26,258
618
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.
 

Related Threads for: Closed set in an open set

Replies
9
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
740
  • Last Post
Replies
8
Views
2K
Replies
2
Views
11K
  • Last Post
Replies
10
Views
2K
Replies
10
Views
3K
  • Last Post
Replies
15
Views
6K
Top