# A generalization of a theorem of Hurewicz for quasi-Polish spaces

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 Π^0_2-subset homeomorphic to one of these four spaces.

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