We identify four countable topological spaces S-2, S-1, S-D, and S-0 which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the T-2, T-1, T-D, and T-0-separation axioms. S-2 is the space of rationals, S-1 is the natural numbers with the cofinite topology, SD is an infinite chain without a top element, and So is the set of finite sequences of natural numbers with the lower topology induced by the prefix ordering. Our main result is a generalization of Hurewicz's theorem showing that a co-analytic subset of a quasi-Polish space is either quasi-Polish or else contains a countable Pi(0)(2)-subset homeomorphic to one of these four spaces.