Diamond on ladder systems and countably metacompact topological spaces
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866911890606653440 |
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| author | Carvalho, Rodrigo Inamdar, Tanmay Rinot, Assaf |
| author_facet | Carvalho, Rodrigo Inamdar, Tanmay Rinot, Assaf |
| contents | The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $Δ$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $Δ$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $κ$ for which $X_L$ is not countably metacompact, in particular, not a $Δ$-space. We prove that an affirmative answer holds for the cardinal $κ=cf(\beth_{ω+1})$. Assuming $\beth_ω=\aleph_ω$, we get an example at a much lower cardinal, namely $κ=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $ω$-bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13367 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Diamond on ladder systems and countably metacompact topological spaces Carvalho, Rodrigo Inamdar, Tanmay Rinot, Assaf Logic General Topology 03E05, 54G20 The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $Δ$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $Δ$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $κ$ for which $X_L$ is not countably metacompact, in particular, not a $Δ$-space. We prove that an affirmative answer holds for the cardinal $κ=cf(\beth_{ω+1})$. Assuming $\beth_ω=\aleph_ω$, we get an example at a much lower cardinal, namely $κ=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $ω$-bounded. |
| title | Diamond on ladder systems and countably metacompact topological spaces |
| topic | Logic General Topology 03E05, 54G20 |
| url | https://arxiv.org/abs/2309.13367 |