  We prove that any `finite-type' component of a stability space of a
triangulated category is contractible. The motivating example of such a
component is the stability space of the Calabi--Yau-$N$ category
$\mathcal{D}(\Gamma_N Q)$ associated to an ADE Dynkin quiver. In addition to
showing that this is contractible we prove that the braid group
$\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular
that the spherical twist group $\operatorname{Br}(\Gamma_N Q)$ is isomorphic to
$\operatorname{Br}(Q)$. This generalises Brav-Thomas' result for the $N=2$
case. Other classes of triangulated categories with finite-type components in
their stability spaces include locally-finite triangulated categories with
finite rank Grothendieck group and discrete derived categories of finite global
dimension.
