Bounded coordinate lengths and normal closures in Polish products

We characterize when the abstract normal closure of an element of a countable product of arbitrary Polish groups admits a Polish group topology with the inherited Borel sets. For h=(hᵢ) in ∏ᵢLᵢ, put Bᵢ=hᵢ^{Lᵢ}(hᵢ⁻¹)^{Lᵢ} and Pᵢ=⟨Bᵢ⟩. The normal closure is Polishable precisely when every Pᵢ is Polishable and, for some integer k, Pᵢ=Bᵢᵏ outside finitely many coordinates. In this case it is the countable cyclic extension (∏ᵢPᵢ)⟨h⟩. Failure of the uniform bound yields a continuous injective reduction of E₁, eventual equality of sequences in 2^ℕ, to the coset relations of both the balanced and ordinary normal closures. The argument applies more generally to symmetric analytic coordinate generating sets. We also give an exact criterion for subgroups defined by uniformly bounded finite Borel coordinate lengths; their Polishable branch has coset relation either trivial or Borel bireducible with E₀. Applications include arbitrary nonabelian abstractly simple Polish factors and compact connected Lie factors. For projective unitary factors, a uniform spectral condition is equivalent to Polishability. Examples distinguish the coordinate obstruction, the uniform-width obstruction, and the cyclic obstruction to closedness.

GitHub repository: eulogica/bounded-lengths-normal-closures