We will discuss some topological aspects of the representation theory of smooth-automorphic forms (i.e., not necessarily $ K_\infty $-finite automorphic forms) on $ G(\mathbb A) $, where $ G $ is a connected reductive group defined over a number field $ F $, and $ \mathbb A $ is the ring of adeles of $ F $. For every ideal $ \mathcal J $ of finite codimension in $ \mathcal Z(\mathfrak g) $, we introduce an LF-topology on the space $ \mathcal A_{\mathcal J}^\infty([G]) $ of $ A_G^{\mathbb R} $-invariant smooth-automorphic forms on $ G(\mathbb A) $ that are annihilated by a power of $ \mathcal J $. With respect to this topology, $ \mathcal A_{\mathcal J}^\infty([G]) $ is a smooth representation of $ G(\mathbb A) $. We show that both the decomposition along the parabolic support and the (finer) decomposition along the cuspidal support of $ K_\infty $-finite automorphic forms transfer in a natural way to our larger setting of smooth-automorphic forms. This is joint work with Harald Grobner.