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Reversibility of the implications PS0 → PS1 → CUC in ZF

Determine whether the implications from PS0 to PS1 and from PS1 to CUC are reversible in ZF; specifically, ascertain whether the Countable Union Theorem (CUC) implies PS1 and whether PS1 implies PS0.

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Background

The authors introduce principles PS0, PS1, and PS2 that formalize P-space existence for large sets. They show PS0 → PS1 → CUC, but the converses are not established. This question concerns whether CUC is sufficient to guarantee PS1, and whether PS1 suffices to guarantee PS0, within ZF. The paper provides partial results and independence information but leaves the full reversibility question open.

References

In Theorem \ref{s7:t4}, we observe that $\mathbf{PS}_0\rightarrow\mathbf{PS}_1\rightarrow\mathbf{CUC}$, where $\mathbf{CUC}$ is the Countable Union Theorem (see Definition \ref{s2forms}(2)). The question whether or not the latter implications are reversible is still open.

— Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice (2510.11935 - Tachtsis et al., 13 Oct 2025) in Section 7, after Theorem s7:t4