3. The topology is too weak, and usually not even first countable, so sequences are not enough to define the topology. Most of these examples are based on a product or pointwise-convergence topology: the weak-* topology on the dual $X^*$ of a Banach space $X$, the spectrum of a $C^*$-algebra, Stone-Čech compactifications, etc. (And actually, in the first case, it is often enough just to look at the unit ball of $X^*$, which is metrizable if $X$ is separable, which in applications it usually is.) As compensation, some corollary of Tychonoff's theorem gives us some compactness, which is probably the only reason we've agreed to put up with such an annoyingly weak topology in the first place.