  We construct and study the space C(\R^d,n) of all partitions of \R^d into n
non-empty open convex regions (n-partitions). A representation on the upper
hemisphere of an n-sphere is used to obtain a metric and thus a topology on
this space. We show that the space of partitions into possibly empty regions
C(\R^d,\le n) yields a compactification with respect to this metric. We also
describe faces and face lattices, combinatorial types, and adjacency graphs for
$n$-partitions, and use these concepts to show that C(\R^d,n) is a union of
elementary semialgebraic sets.
