Connectedness of the set having a fixed distance from a closed set 2

Question

This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $mathbb{R}^n$ ($n>1$). Suppose the complement  of $F$ is connected and let
$$A={xin mathbb{R}^n: dist(x, F)=delta}, $$ where $delta>0$ is fixed, and $dist$ is the Euclidean distance.  If $F$ is unbounded with empty interior,  can the complement of $A$  still have a bounded component?

Anthony Quas · Accepted Answer

Here is a picture of the set $F$ (in red) in my comment above (the black lines represent the sphere of radius 2). There is a component of $A$ inside the sphere and a component outside the sphere.

Connectedness of the set having a fixed distance from a closed set 2

One Answer

Add your own answers!

Ask a Question