The author introduces the concept of an \(H\)-chain set \(\Omega\) in a doubling space \(X\); roughly speaking this means that there exists a ``fairly short'' chain of balls from any \(x\in\Omega\) to a fixed \(x_0\in\Omega\). \(H\)-chain sets generalize the notion of Hölder domains in Euclidean space but are not necessarily connected. It is shown that every \(H\)-chain set \(\Omega\) is mean porous and that its outer layer has measure bounded by a power of its thickness. As a consequence the author shows that a John-Nirenberg type inequality holds on an open subset \(\Omega\) of a doubling space \(X\) if, and often only if, \(\Omega\) is an \(H\)-chain set.