I'm reading A. Borel's "Linear Algebraic Groups".
At an early point in the book, the author establishes the following concepts:
(Let $K$ be an algebraically closed field, and $k$ a subfield of $K$)
- $K$-space: a topological space $X$ along with a sheaf $O$ of $K$-algebras on $X$ such that the stalks are local rings
- affine $K$-scheme: a $K$-space of the form Spec$_K(A)$;
(Spec$_K(A)$ denotes all the maximal ideals of a $K$-algebra $A$) - $K$-scheme: a $K$-space which is locally isomorphic to an affine $K$-scheme
He then proceeds to introduce the concept of a $k$-structure on a $K$-scheme $(X, O)$, which consists of the following:
a) a $k$-topology on X (which is weaker than the original topology on $X$);
b) a $k$-structure on each of the $O(U)$, where $U$ is $k$-open.
The question is: what does a "$k$-topology on $X$" mean, and how do we define one? Any examples are also welcome :)