$Ψ$-Spaces and Semi-Proximality
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916543375343616 |
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| author | Almontashery, Khulod Rodrigues, Vinicius de Oliveira Szeptycki, Paul J. |
| author_facet | Almontashery, Khulod Rodrigues, Vinicius de Oliveira Szeptycki, Paul J. |
| contents | We discuss the proximal game and semi-proximality in $Ψ$-spaces of almost disjoint families over an infinite countable set and $Ψ$-spaces of ladder systems on $ω_1$. We show that a semi-proximal almost disjoint families must be nowhere MAD, anti-Luzin and characterize semi-proximality for a class of ${\mathbb R}$-embeddable almost disjoint families. We show that a $Ψ$-spaces defined from a uniformizable ladder system is semi-proximal and a $Ψ$-space defined on a $\clubsuit^*$ sequence is not semi-proximal. Thus the existence of non-semi-proximal $Ψ$-space over a ladder system is independent of ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $Ψ$-Spaces and Semi-Proximality Almontashery, Khulod Rodrigues, Vinicius de Oliveira Szeptycki, Paul J. General Topology Logic Primary: 54D15, 54D80 Secondary: 03E75, 03E05, 54D15, 54D80 We discuss the proximal game and semi-proximality in $Ψ$-spaces of almost disjoint families over an infinite countable set and $Ψ$-spaces of ladder systems on $ω_1$. We show that a semi-proximal almost disjoint families must be nowhere MAD, anti-Luzin and characterize semi-proximality for a class of ${\mathbb R}$-embeddable almost disjoint families. We show that a $Ψ$-spaces defined from a uniformizable ladder system is semi-proximal and a $Ψ$-space defined on a $\clubsuit^*$ sequence is not semi-proximal. Thus the existence of non-semi-proximal $Ψ$-space over a ladder system is independent of ZFC. |
| title | $Ψ$-Spaces and Semi-Proximality |
| topic | General Topology Logic Primary: 54D15, 54D80 Secondary: 03E75, 03E05, 54D15, 54D80 |
| url | https://arxiv.org/abs/2412.18982 |