DISTINGUISHING ENDPOINT SETS FROM ERDŐS SPACE

The paper proves that the escaping endpoint set Ė(f) for f(z)=e^z−1 is not homeomorphic to Q×X for any space X via a brush-model plus a topological lemma and an explicit simple-closed-curve construction around endpoints; the argument is coherent and complete in the provided text (see the statement of Theorem 1 and its proof outline and completion) . By contrast, the model’s proof hinges on unproven—and likely false—claims: (i) that the escaping endpoints Ė(f) are homeomorphic to an open subset of Z^N (hence Polish/Baire), and (ii) that the address–endpoint map is a homeomorphism on the subspace of addresses with positive minimal potential. The paper’s brush model does not imply these topological identifications; in particular, the set Σ_fast={s:t_s>0} need not be open in the product topology on Z^N, and “escaping endpoint = t_s>0” is not justified. Consequently, the model’s Baire-vs-meagre argument lacks the required foundations.