While defining affine and projective varieties we consider Zariski topology on $\mathbb A^n$ and $\mathbb P^n$. In the process we define $ Z(T)$, zero set of $T $ where $T\subset A=k[x_1,...,x_n]$ and Ideal of any subset $Y \subset \mathbb A^n$. Now, we define $Z(\emptyset)= \mathbb A^n$ and $I(\emptyset)= A$. Why we choose such a definition for them? I am convinced that it follows vacuously from the definition given in Hartshorne , which says , $Z(T)= $ set of all points in $\mathbb A^n$ where all polynomials in $T $ vanish. And $I(Y)=$ set of all polynomials in $A$ which vanishes at all points of $Y$. But my question is any other description also satisfies vacuously here . Why we choose this particular one.
Similarly I am confused for the case of projective varieties too. Need some help in understanding the motivation behind this.
There are many reasons, but I guess what I would think as the most important one is that it is the only definition for which $S\subseteq T \implies Z(T)\subseteq Z(S)$ (and the obvious similar statement for $I$), which are obviously true for nontrivial ideals, to hold.