I am struggling to understand some of the basic definitions from Borel's book on linear algebraic groups.
Let $ K/k $ be a field extension, Borel defines a $k$-structure of a $K$-scheme as follows:
Question 1: Is this equivalent to saying that $X = X' \times_k \operatorname{spec}K$ for some $k$-scheme $X'$?
Next, he defines the rational points:
While it is first defined for $K$-algebras, it seems like we are mostly interested in subfields of $K$, for which I am not sure I understand the definition. After reading another MSE post, and hoping that my understanding of the previous definition is correct,
Question 2: Is it true that $V(k')$ is defined to be $V'(k')$, where $V = V' \times_k \operatorname{spec}K$? And is $V(K)$ just the set underlying $V$?
And finally, Question 3: If $Z\subset V$ is closed (or open) in the $k$-topology, does $Z\subset V(k)$? Why?