Intersections of maximal staircase sets

A compact set is staircase connected if every two points a, b ∈ S can be connected by an x-monotone and y-monotone polygonal path with sides parallel to the coordinate axes. In [5] we have introduced the concepts of staircase k-stars and kernels. In this paper we prove that if the staircase k-kernel is not empty, then it can be expressed as the intersection of a covering family of maximal subsets of staircase diameter k of S.