Are zero sets of polynomial equations closed because of the fundamental theorem of algebra?
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Tu Manifolds Section 15.2 Tu Manifolds Section 9.2 Level sets in Tu Manifolds does not look different from fibres in Artin Algebra or level curves in Stewart Calculus So I think I understand correctly when I say that I think that: Zero sets of polynomial equations (I think Tu should say polynomial functions) over smooth manifolds are closed because by the fundamental theorem of algebra, zero sets are finite, and finite sets are closed because smooth manifolds are Hausdorff. Is that correct? Can we generalize? Level sets of polynomial equations (I think Tu should say polynomial functions) over topological manifolds are closed because by the fundamental theorem of algebra, level sets are finite, and finite sets are closed because topological manifolds are Hausdorff. Is this altern