The definition of a $\sigma$-field $\mathscr{F}$ on a set $X$ (or $\sigma$-ring) requires $\mathscr{F}$ to be a non-empty subset of $\mathscr{P}(X)$. Why is this convention taken? What is the issue with allowing for empty measurable spaces?
Similarly, why is a topology defined to be a non-empty collection of $\mathscr{P}(X)$? Whats the issue with allowing an "empty" topological space?