ON HAUSDORFF COVERS FOR NON-HAUSDORFF GROUPOIDS

The paper’s Theorem 3.13 (and its formulation as Theorem D) exactly characterizes the images of i: C_c(G) → C_c(Ĝ) and, in the ample case, i': R[G] → R[Ĝ] as the decomposable functions, with a full proof relying on the Fell-topology construction of the Hausdorff cover and auxiliary lemmas (e.g., Lemma 3.12) that ensure finite summations and continuity on Ĝ. This is explicitly stated and proved in the uploaded paper , and summarized up front as Theorem D . By contrast, the candidate solution makes a critical incorrect assumption that the canonical inclusion ]: G → Ĝ is continuous; the paper explicitly notes that ] is generally non-continuous (though with dense image) . The candidate’s backward direction hinges on composing a continuous f~ with ] to produce a continuous f on G, which is unjustified without continuity of ]. The candidate also mismatches domains (using L^2(G) instead of C_c(G)) and similarly assumes local constancy in the ample case from the same false continuity premise. Although some lemmas in the candidate (e.g., |g ∩ U| ≤ 1 for Hausdorff open bisections U, and finiteness of g ∩ K for compact K) align with the paper’s framework (open bisections form a Hausdorff basis and the Fell description of Ĝ ), these do not repair the main gap. Net: the paper’s result and proof are correct and complete; the model’s proof is flawed in key steps that depend on continuity of ].