In the first chapter of his book "Linear Algebraic Groups", Springer considers a situation where $k$ is an algebraically closed field and $F$ is a subfield, and seeks to define the notion of an $F$-structure on an affine $k$-variety $X$.
He begins by defining the notion of an $F$-closed subset of $X$. In Humphreys we have the notion of an $F$-closed set and a set which is defined over $F$, the former being an algebraic set which is defined by an ideal which is defined over $F$, and the latter being an algebraic set with the property that the corresponding ideal is defined over $F$. These two notions may be distinct as there could be an ideal which is defined over $F$ whose radical is not defined over $F$.
Going by what Springer writes at first in Section 1.3.8 he seems to have in mind the second notion, the notion of an algebraic set with the property that the corresponding ideal is defined over $F$. But then I wonder how we know that these algebraic sets are the closed sets of a topology, and also Springer goes on to say that an algebraic set defined by a single polynomial with coefficients in $F$ would have this property, but this is not clear because the radical of the ideal generated by the polynomial may not be defined over $F$.
So perhaps there is an error in Springer's exposition? What would be the best way of fixing it up?