Exercise 2, Appendix 1

Assume that V is the product of two vector spaces $V_1$ and $V_2$.

a) The Zariski topology on V is finer than the product topology of the Zariski topologies on $V_1$ and $V_2$; it is strictly finer if $V_1\not= 0$ and $V_2\not= 0$.

b) If $A_1$ (resp. $A_2$) is a subset of $V_1$ (resp. $V_2$), the closure of $A_1\times A_2$ is the product of the closures of $A_1$ and $A_2$.