  We prove that every open subset of a euclidean building is a finite
dimensional absolute neighborhood retract. This implies in particular that such
a set has the homotopy type of a finite dimensional simplicial complex. We also
include a proof for the rigidity of homeomorphisms of euclidean buildings. A
key step in our approach to this result is the following: the space of
directions $\Sigma_oX$ of a CAT$(\kappa)$ space $X$ is homotopy quivalent to a
small punctured disk $B_\eps(X,o)\setminus o$. The second ingredient is the
local homology sheaf of $X$. Along the way, we prove some results about the
local structure of CAT$(\kappa)$-spaces which may be of independent interest.
