Y is a map. Y is continuous. Hence f is continuous. It is a Cauchy-complete geometric site. We prove that limits and colimits exist in k - Space .
Y are morphisms of k -spaces.
Limits and colimits exist in the category of k-spaces. It remains to prove that fibre coproducts exist in k - Space . Y are two morphisms of k -spaces. The fibre product of separated spaces is separated. We now use the notion of k -spaces to define an algebraic variety in the sense of Serre.