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On Some Open Problems about P-Spaces, Strongly Quasi Baire Spaces and Choice

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Abstract

In set theory without the full power of the axiom of choice (\(\textbf{AC}\)), we resolve open problems from Keremedis, Olfati and Wajch “On P-spaces and \(G_{\delta }\)-sets in the absence of the Axiom of Choice” on the deductive strength of statements concerning P-spaces and strongly quasi Baire spaces via positive and independence results. For some of the independence results, we construct three new permutation models of \(\textbf{ZFA}+\lnot \textbf{AC}\), where \(\textbf{ZFA}\) denotes the Zermelo–Fraenkel set theory with atoms. Part of our \(\textbf{ZFA}\)-independence proofs are transferable to \(\textbf{ZF}\) (i.e. Zermelo–Fraenkel set theory without \(\textbf{AC}\)).

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Notes

  1. \(\textbf{PCACCLO}\) was introduced in [8].

  2. This implication was communicated by me to the authors of [8].

  3. We note that the proof of (1) is fairly similar to the proof of (2); work again by way of contradiction, and replace, in the corresponding definition of \(\mathcal {B}_{i}\), “\(F\in [A_{i}]^{\le \omega }\)” by “\(F\in [A_{i}]^{<\omega }\)”.

  4. In [12, Lemma 1] it was shown that \(\textbf{P}_{i}=\bigcup \mathcal {Z}_{i}\), and thus \(\textbf{P}_{i}\) is cuf in \(\mathcal {N}\).

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Acknowledgements

The author is most grateful to the anonymous referee for several comments and suggestions, which lead to an improvement of the quality of the paper.

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  1. Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi, 83200, Samos, Greece

    Eleftherios Tachtsis

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Tachtsis, E. On Some Open Problems about P-Spaces, Strongly Quasi Baire Spaces and Choice. Results Math 79, 19 (2024). https://doi.org/10.1007/s00025-023-02049-4

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