  We show that any filtering family of closed convex subsets of a
finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual
bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results
known for proper CAT(0) spaces may be extended to finite-dimensional spaces,
including the existence of canonical fixed points at infinity for parabolic
isometries, algebraic and geometric restrictions on amenable group actions, and
geometric superrigidity for non-elementary actions of irreducible uniform
lattices in products of locally compact groups.
