A subspace $S$ of a topological space $X$ is said to separate

if $X - S$ contains more than one path-component. The classical Alexander

duality theorem implies that if a subset $A$ of the $n$-dimensional sphere

separates it, then $A$ must be of (topological) dimension $n - 1$. Coarse

separation is an analogue of topological separation in the world of

metric spaces. Coarse separation arises naturally in geometric group

theory. I will introduce asymptotic dimension which is an analogue of

topological dimension in the coarse setting. It was conjectured that

every coarsely separating subset of $\mathbb{R}^n$ is of asymptotic dimension at

least $n-1$. I have proved this conjecture. I will give an outline of

the proof of this result.