Skeletons and Central Sets
DH Fremlin · Proceedings of the London Mathematical Society · 1997
Let Ω be an open proper subset of Rn. Its skeleton is the set of points with more than one nearest neighbour in the complement of Ω its central set is the set of centres in maximal open balls included in Ω. Intuitively, if we think of Ω as a land mass in which height is proportional to distance from the sea, its skeleton and central set can be thought of as corresponding to ridges in the mountains of Ω. In this note I discuss the metric and topological properties of such sets. I show that any skeleton in Rn is Fσ, and has dimension at most n − 1, by any of the usual measures of dimension; that if Ω is bounded and connected, its skeleton and central set are connected; and that Ω separates Rn iff its skeleton does iff its central set does. Any central set in Rn is a Gδ set of topological dimension at most n − 1. In the plane, I show that both skeletons and central sets are locally path-connected, and indeed include many paths of finite length. For any Ω, its central set includes its skeleton; I give examples to show that the central set can be significantly larger than the skeleton. 1991 Mathematics Subject Classification: 54F99.