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.

Source : oai:arXiv.org:1711.09326

DOI : 10.23638/LMCS-14(1:13)2018

Volume: Volume 14, Issue 1

Published on: February 6, 2018

Submitted on: November 29, 2017

Keywords: Mathematics - General Topology,Computer Science - Logic in Computer Science

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