Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917011279314944 |
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| author | Tachtsis, Eleftherios Wajch, Eliza |
| author_facet | Tachtsis, Eleftherios Wajch, Eliza |
| contents | For an infinite set $X$, a closed under finite unions family $\mathcal{Z}$ with $[X]^{<ω}\subseteq\mathcal{Z}\subseteq\mathcal{P}(X)$, and any $\mathcal{A}\subseteq\mathcal{P}(X)$, the topology $τ_{\mathcal{A}}[\mathcal{Z}]=\{V\in\mathcal{A}: (\forall x\in V)(\exists z\in \mathcal{Z})(x\cap z=\emptyset \wedge \{y\in\mathcal{A}: x\subseteq y\subseteq X\setminus z\}\subseteq V)\}$ on $\mathcal{A}$ is investigated to give answers to the following open problem in various models of $\mathbf{ZF}$ or $\mathbf{ZFA}$: Is there a non-empty Hausdorff, crowded zero-dimensional $P$-space in the absence of the axiom of choice? Spaces of the form $\mathbf{S}(X, [X]^{\leqω})=\langle \mathcal{A}, τ_{\mathcal{A}}[\mathcal{Z}]\rangle$ for $\mathcal{A}=[X]^{<ω}$ and $\mathcal{Z}=[X]^{\leqω}$ are of special importance here. Among many other results, the following theorems are proved in $\mathbf{ZF}$: (1) If $X$ is uncountable, then $\mathbf{S}(X, [X]^{\leqω})$ is a crowded zero-dimensional Hausdorff space, and if $X$ is also quasi Dedekind-finite, then $\mathbf{S}(X, [X]^{\leqω})$ is a $P$-space; (2) $\mathbf{S}(ω_1, [ω_1]^{\leqω})$ is a $P$-space if and only if $ω_1$ is regular; (3) the axiom of countable choice for families of finite sets is equivalent to the statement ``for every infinite Dedekind-finite set $X$, $\mathbf{S}(X,[X]^{\leqω})$ is a $P$-space''; (4) the statement ``$\mathbb{R}$ admits a topology $τ$ such that $\langle\mathbb{R}, τ\rangle$ is a crowded, zero-dimensional Hausdorff $P$-space'' is strictly weaker than the axiom of countable choice for families of subsets of $\mathbb{R}$; (5) the statement ``there exists a non-empty, well-orderable crowded zero-dimensional Hausdorff $P$-space'' is strictly weaker than ``$ω_1$ is regular''. A lot of relevant independence results are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice Tachtsis, Eleftherios Wajch, Eliza General Topology Logic 54G10, 03E25, 03E35 For an infinite set $X$, a closed under finite unions family $\mathcal{Z}$ with $[X]^{<ω}\subseteq\mathcal{Z}\subseteq\mathcal{P}(X)$, and any $\mathcal{A}\subseteq\mathcal{P}(X)$, the topology $τ_{\mathcal{A}}[\mathcal{Z}]=\{V\in\mathcal{A}: (\forall x\in V)(\exists z\in \mathcal{Z})(x\cap z=\emptyset \wedge \{y\in\mathcal{A}: x\subseteq y\subseteq X\setminus z\}\subseteq V)\}$ on $\mathcal{A}$ is investigated to give answers to the following open problem in various models of $\mathbf{ZF}$ or $\mathbf{ZFA}$: Is there a non-empty Hausdorff, crowded zero-dimensional $P$-space in the absence of the axiom of choice? Spaces of the form $\mathbf{S}(X, [X]^{\leqω})=\langle \mathcal{A}, τ_{\mathcal{A}}[\mathcal{Z}]\rangle$ for $\mathcal{A}=[X]^{<ω}$ and $\mathcal{Z}=[X]^{\leqω}$ are of special importance here. Among many other results, the following theorems are proved in $\mathbf{ZF}$: (1) If $X$ is uncountable, then $\mathbf{S}(X, [X]^{\leqω})$ is a crowded zero-dimensional Hausdorff space, and if $X$ is also quasi Dedekind-finite, then $\mathbf{S}(X, [X]^{\leqω})$ is a $P$-space; (2) $\mathbf{S}(ω_1, [ω_1]^{\leqω})$ is a $P$-space if and only if $ω_1$ is regular; (3) the axiom of countable choice for families of finite sets is equivalent to the statement ``for every infinite Dedekind-finite set $X$, $\mathbf{S}(X,[X]^{\leqω})$ is a $P$-space''; (4) the statement ``$\mathbb{R}$ admits a topology $τ$ such that $\langle\mathbb{R}, τ\rangle$ is a crowded, zero-dimensional Hausdorff $P$-space'' is strictly weaker than the axiom of countable choice for families of subsets of $\mathbb{R}$; (5) the statement ``there exists a non-empty, well-orderable crowded zero-dimensional Hausdorff $P$-space'' is strictly weaker than ``$ω_1$ is regular''. A lot of relevant independence results are obtained. |
| title | Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice |
| topic | General Topology Logic 54G10, 03E25, 03E35 |
| url | https://arxiv.org/abs/2510.11935 |