A note on systems of rectangular islands the continuous case - Lengvárszky Zsolt (Acta scientiarum mathematicarum, 2011)
A real-valued height function / is defined on a closed rectangle R. A rectangle S C R is an /-island if there exists an open set G C R containing S such that f{x) < infs f for every x 6 G\S. The set of all /-islands is called a system of (rectangular) islands. In this paper we prove that there exists a maximal system of islands of cardinality No> and that the size of a maximal system of islands is either countable or continuum.