Say that $V$ is an affine variety defined over $k$ and that $W$ is a locally closed subset of $V$. Is there some sufficient condition for $W$ to have a canonical $k$-structure?



making sense of a locally closed set being defined over k
I believe that $W$ has a canonical $k$-variety structure without our imposing any additional assumptions. 


The closure $\overline{W}$ has a canonical affine $k$-variety structure since it is a closed subset of $V$. By virtue of being locally closed, $W$ is open in $\overline{W}$. Therefore, $W$ has the structure of an open subvariety of $\overline{W}$. 


In general, a locally closed subset of an affine variety is called a quasi-affine variety.


