Topological noetherianity for powers of algebraic representations

Powers of a polynomial $\operatorname{GL}$-representation are topologically Noetherian under the action of $\operatorname{Sym} \times \operatorname{GL}$. We extend this result to powers of algebraic representations of the orthogonal and the symplectic groups, proving topological Noetherianity under the action of $\operatorname{Sym} \times \operatorname{O}$ and $\operatorname{Sym} \times \operatorname{Sp}$ respectively. This work builds on arXiv:2212.05790 and arXiv:1708.06420, and it provides further evidence that infinite powers of topologically Noetherian varieties remain topologically Noetherian up to permutations.