THE TOPOLOGICAL STRUCTURE OF FUNCTION SPACE OF TRANSITIVE MAPS

The paper proves that T(I) and its closure T̄(I) in C(I) are each homeomorphic to ℓ2 by establishing: (i) an extension property (Statement E) to get ANR/AR structure and that T(I) is homotopy dense in T̄(I), hence T̄(I) is also an AR; (ii) SDAP for T(I) via the Kolyada–Misiurewicz–Snoha box-map homotopies; and (iii) topological completeness (T̄(I) closed; T(I) is Gδ in C(I)); thereby applying Toruńczyk’s characterization of ℓ2 to both spaces . The candidate solution follows the same plan and arrives at the same conclusion. Two minor issues: (A) the paper’s passage “hence [T̄(I)] also has SDAP” is asserted after proving SDAP for T(I), but the direction of the cited inheritance result (HD subspace of an ANR with SDAP has SDAP) goes the other way; a short homotopy-pushing argument fills this gap and yields SDAP for T̄(I) as well . (B) the model’s remark that the identity map cannot be uniformly approximated by transitive maps is false; the paper gives explicit transitive maps converging to the identity . These do not affect the main theorem. Overall, the proofs coincide in structure and are correct up to the noted, easily rectified omissions.