topological spaces, and whose nonlogical primitives are interpreted as topological relations and functions taking these subsets as arguments. Thus, topological constraint languages typically allow us to make assertions such as "region V1 touches the boundary of region V2", "region V3 is connected" or "region V4 is a proper part of the closure of region V5". A formula in a topological constraint language is said to be satisfiable if there exists an assignment to its variables of regions from some topological space under which is made true. This paper introduces a topological constraint language which, in addition to the usual mechanisms for expressing Boolean combinations of regions and their topological closures, includes primitives for bounding the number of components of a region. We call this language T CC, a rough acronym for "topological constraint language with component counting". (Thus, T CC extends earlier topological constraint languages based on the so-called RCC-primitives.) Our main result is that the problem of determining the satisfiability of a T CC-formula is NEXPTIME-complete.