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

[ehrenfest]

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

 

[CompuChip]

Why would the distance have to be infinitesimal? If I have a closed \epsilon-ball, then wouldn't an open \frac{\epsilon}{2}-ball do?
Now generalize to any closed set.

 

[HallsofIvy]

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.

 

[ehrenfest]

Compuchip is right about what I am asking.
So, does anyone know how to do that?

 

[ehrenfest]

anyone?

 

[Dick]

You need more than just closed, is it compact?

 

[ehrenfest]

Yes. It is closed an bounded. How do you use compactness here?

 

[Dick]

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.

 

[ehrenfest]

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?

 

[Dick]

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.