Abstract. A real-valued height function $f$ is defined on a closed rectangle $R$. A rectangle $S\subset R$ is an $f$-island if there exists an open set $G\subset R$ containing $S$ such that $f(x)< \inf_{S} f$ for every $x\in G\setminus S$. The set of all $f$-islands is called a {\it system of (rectangular) islands.} In this paper we prove that there exists a maximal system of islands of cardinality $\aleph_0$, and that the size of a maximal system of islands is either countable or continuum.

AMS Subject Classification (1991): 05A05, 54A25

Keyword(s): maximal systems of islands, countable, continuum, laminar system

Received September 7, 2009, and in final form March 3, 2010. (Registered under 161/2009.)