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comment Commutator of algebraic subgroups is connected
Chevalley's theorem on preservation of constructibility under images pervades the beginning of the theory of algebraic groups. The constructible subsets of a noetherian topological space are the finite unions of locally closed sets. Check Wikipedia under "constructible set" (though the example there is weak, since it is even locally closed; a better example is the subset of the plane given by the union of the origin and the complement of the $x$-axis). By the way, the end of your posted question made me think that you were well aware that algebraic groups aren't topological groups.
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answered Commutator of algebraic subgroups is connected