In "Infinite dimensional analysis, A hitchhikers guide" by Aliprantis and Border, they write that these 2 classes of topologies "by and large include everything of interest".
@Pete Clarke: I was asking if it holds for mathematics in general, and not just for functional analysis. From the answers I get the impression that their statement is a fairly good first approximation.
@Gerald Edgar: I thought I read in Mathoverflow that the Zariski topology can be regarded as a weak topology (at least in some cases).
