arXiv · 1904.11168
For every uncountable regular $κ$, we give two examples of proper posets which turn improper in some $κ$-closed forcing extension.
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Yasuo Yoshinobu. 2019-08-02. Properness under closed forcing. https://arxiv.org/abs/1904.11168
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We study the Borel and analytic subsets of the spaces ${}^κκ$ and ${}^κ2$ endowed with ideal topologies, where $κ$ is a regular uncountable cardinal, thereby addressing some open problems of the literature. We provide a systematic analysis of the Borel hierarchy for an arbitrary ideal topology. In particular, we formulate a sufficient condition ensuring that the hierarchy does not collapse, demonstrate that every Borel set in such a topology is analytic, and establish the existence of a set that is not Borel. Our main result shows that, when the underlying ideal contains an unbounded subset, the collection of analytic sets coincides with the full power set of the ambient space. Finally, we prove that the Approximation Lemma holds in the setting of ideal topologies.
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