Let $X$ be a topological space (or more specifically, $\mathbb{C}^N$ endowed with the Zariski topology), and let $S \subseteq X$ be a constructible set, i.e. $S=\cup_{i=1}^n C_i \cap U_i$, where the $C_i$'s are closed in $X$ and the $U_i$'s are open. Then $S$ contains an open dense subset of its closure $\overline{S}$ (this is true even if $\overline{S}$ is not irreducible-- see Lemma 2.1 in this paper). I would like a list of some known conditions that imply $S$ is itself an open dense subset of its closure (also known as *locally closed*).

As one example, if there is a transitive topological group action on $S$, then $S$ is locally closed.

One might hope that if $S$ is constructible, irreducible, and connected, then it is locally closed. But this is contradicted by the image of the map on $\mathbb{C}^2$ given by $(x,y)\mapsto(x,xy)$, which satisfies all of these properties, but is not locally closed.

One personal motivation for this question is I am trying to prove that the set of tensors in $\mathbb{C}^{N_1}\otimes\dots\otimes \mathbb{C}^{N_d}$ of tensor rank at most $r$ is locally closed (i.e., an open dense subset of its closure, the set of tensors of border rank at most $r$). I have seen this fact mentioned several times, but have not been able to prove it.

border rankat most $r$, or equivalently it is the $r$-th secant to the Segre variety). The set of tensors of tensor rank at most $r$ is easily shown to be constructible, though. I want to prove that it is locally closed. $\endgroup$