For each a ∈ C let fa be defined by z → e z + a, and let F (fa) denote the Fatou set of fa. In this paper we prove that the meandering Julia set Jm(fa) is homeomorphic to the space of irrationals P whenever a ∈ F (fa), extending recent results by Vasiliki Evdoridou and Lasse Rempe-Gillen. It follows that the radial Julia set Jr(fa) has topological dimension zero for all attracting and parabolic parameters, including all a ∈ (−∞, −1). This has several consequences for the topologies of the escaping and fast escaping sets and their endpoints.