The discussion revolves around finding the minimum epsilon radius for a closed set situated within an open set in a metric space. Participants are exploring the relationship between the boundaries of the closed and open sets, particularly focusing on the concept of distance and its implications in this context.
The conversation is ongoing, with various interpretations being explored. Some participants have suggested potential approaches involving neighborhoods and compactness, while others have raised concerns about the validity of these methods and the assumptions being made. No consensus has been reached yet.
There is a mention of the closed set being compact, which adds a layer of complexity to the discussion. Participants are also grappling with the implications of using finite subcovers and the nature of the boundaries involved.
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?