一句话：识别四个对应 T2/T1/TD/T0 的非 quasi-Polish 典范可数空间，并证明 co-analytic 子集的 Hurewicz 型二分定理。
本文研究 quasi-Polish 空间的刻画与 Hurewicz 定理的推广。作者首先给出四个可数拓扑空间 S₂、S₁、S_D、S₀，分别对应 T₂、T₁、T_D、T₀ 分离公理，并作为非 quasi-Polish 空间的典范例子：S₂ 为有理数空间，S₁ 为带余有限拓扑的自然数集，S_D 为无顶元的无限链，S₀ 为自然数有限序列以前缀序诱导的下拓扑。主要结果是 Hurewicz 定理的推广：quasi-Polish 空间的 co-analytic 子集要么本身是 quasi-Polish，要么包含一个可数的 Π⁰₂ 子集，且该子集同胚于上述四个空间之一。证明采用描述集合论与拓扑方法。
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, $S_D$ is an infinite chain without a top element, and $S_0$ 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.