Topological-Algebraic Collapse of Equational Theories via Finite Continuous Quotients

For a topological algebra A, we define a profinite equational theory Th_pf(A) by testing ordinary identities on all finite continuous quotients of A. We show that Th_pf(A) coincides with the ordinary equational theory of the profinite completion Â, and that Th_alg(A) ⊆ Th_pf(A) holds for every A. As consequences, collapse Th_pf(A) = Th_alg(A) holds for residually finite discrete algebras and, more generally, for profinite topological algebras. For compact Hausdorff algebras, we prove that Stone plus residual finiteness implies profiniteness and hence collapse. We also identify clopen subsets of  with recognizable subsets of A and exhibit the circle group as an example where Th_alg(T) ⊊ Th_pf(T).