  This paper is about geometric and topological properties of a proper CAT(0)
space $X$ which is cocompact - i.e. which has a compact generating domain with
respect to the full isometry group. It is shown that geodesic segments in $X$
can "almost" be extended to geodesic rays. A basic ingredient of the proof of
this geometric statement is the topological theorem that there is a top
dimension $d$ in which the compactly supported integral cohomology of $X$ is
non-zero. It is also proved that the boundary-at-infinity of $X$ (with the cone
topology) has Lebesgue covering dimension $d-1$. It is not assumed that there
is any cocompact discrete subgroup of the isometry group of $X$; however, a
corollary for that case is that "the dimension of the boundary" is a quasi-
isometry invariant of CAT(0) groups. (By contrast, it is known that the
topological type of the boundary is not unique for a CAT(0) group.)
