Every strict affine ind-variety has a non-countably-generated coordinate ring, and every affine ind-variety admits a countable dense space of regular functions.
Pro-affine Algebras, Ind-Affine groups and the Jacobian Problem
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On the generators of coordinate algebras of affine ind-varieties
Every strict affine ind-variety has a non-countably-generated coordinate ring, and every affine ind-variety admits a countable dense space of regular functions.