Constructions of crowded zero-dimensional Hausdorff P-spaces without the Axiom of Choice

Eliza Wajch
Institute of Mathematics, University of Siedlce, 3 Maja 54, 08-110 Siedlce, Poland

The results presented here form part of the author’s joint work with Eleftherios Tachtsis [2]. They are motivated by the question posed in [1]: is the existence of a non-discrete Tychonoff P-space provable in ZF? A topological space whose every G_δ-set is open is called a P-space.

Throughout, our set-theoretic framework is ZF or ZFA. Among several related results, we show that non-empty zero-dimensional crowded Hausdorff P-spaces exist in every permutation model of ZFA and in every model of ZF having an aleph of uncountable cofinality. A key ingredient in our analysis is the following construction of zero-dimensional Hausdorff spaces which, under additional hypotheses, yields P-spaces.

Let X be an infinite set, and let 𝒵 be a family of subsets of X closed under finite unions and containing [X]^<ω. For x ∈ [X]^<ω and z ∈ 𝒵 with x ∩ z = ∅, define B(x,z) = {y ∈ [X]^<ω : x ⊆ y ⊆ X \ z}. Let 𝒯 be the topology on [X]^<ω such that, for each x ∈ [X]^<ω, the family {B(x,z): z∈ 𝒵 and x ∩ z = ∅} forms a neighborhood base at x. The resulting space S(X, 𝒵) = ([X]^<ω , 𝒯) is Hausdorff and zero-dimensional, and it is crowded whenever X is not a member of 𝒵. Assuming that 𝒵 is a bornology on X, we obtain that the space S(X, 𝒵) is homogeneous, and if it is a P-space, then 𝒵 is a σ-ideal. In particular, S(X, [X]^<ω) is a P-space if and only if the set X is quasi Dedekind-finite. The space S(ω₁, [ω₁]^≤ω) is a P-space if and only if ω₁ has uncountable cofinality. Every denumerable family of non-empty finite sets admits a choice function if and only if, for every infinite set X, either X is Dedekind-infinite or S(X, [X]^≤ω) is a P-space. If every denumerable family of non-empty subsets of the real line ℝ has a choice function, then there exists a topology 𝒯 on ℝ such that (ℝ 𝒯) is a zero-dimensional, crowded Hausdorff P-space. The converse implication is false in the Basic Cohen Model of ZF. In fact, in the Basic Cohen Model, for every infinite set X, the space S(X, [X]^≤ω) is a P-space.

References

• [1] K. Keremedis, A. R. Olfati, and E. Wajch, On P-spaces and G_δ-sets in the absence of the axiom of choice, Bull. Belg. Math. Soc. Simon Stevin 30 (2), 194–236 (2023).
• [2] E. Tachtsis and E. Wajch, Constructing crowded Hausdorff P-spaces in set theory without the axiom of choice, submitted manuscript (2025), https://arxiv.org/abs/2510.11935