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In this paper we describe several new aspects of the foundations of therepresentation theory of the space of smooth-automorphic forms (i.e., notnecessarily $K_\infty$-finite automorphic forms) for general connectedreductive groups over number fields. Our role model for this space ofsmooth-automorphic forms is a ''smooth version'' of the space of automorphicforms, whose internal structure was the topic of a famous paper of Franke. Weprove that the important decomposition along the parabolic support, and theeven finer - and structurally more important - decomposition along the cuspidalsupport of automorphic forms transfer in a topologized version to the largersetting of smooth-automorphic forms. In this way, we establishsmooth-automorphic versions of the main results of a paper of Franke-Schwermerand of Moeglin-Waldspurger's book, III.2.6.

On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations