Weight, net weight, and elementary submodels
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912294581043200 |
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| author | Dow, Alan Juhász, István |
| author_facet | Dow, Alan Juhász, István |
| contents | In this note we prove several theorems that are related to some results and problems from [6]. We answer two of the main problems that were raised in [6]. First we give a ZFC example of a Hausdorff space in $C(ω_1)$ that has uncountable net weight. Then we prove that after adding any number of Cohen reals to a model of CH, in the extension every regular space in $C(ω_1)$ has countable net weight. We prove in ZFC that for any regular topology of uncountable weight on $ω_1$ there is a non-stationary subset that has uncountable weight as well. Moreover, if all final segments of $ω_1$ have uncountable weight then the assumption of regularity can be dropped. By [6], the analogous statements for the net weight are independent from ZFC. Our proofs of all these results make essential use of elementary submodels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weight, net weight, and elementary submodels Dow, Alan Juhász, István Logic General Topology 54A25, 03E05, 03E10, 03E35 In this note we prove several theorems that are related to some results and problems from [6]. We answer two of the main problems that were raised in [6]. First we give a ZFC example of a Hausdorff space in $C(ω_1)$ that has uncountable net weight. Then we prove that after adding any number of Cohen reals to a model of CH, in the extension every regular space in $C(ω_1)$ has countable net weight. We prove in ZFC that for any regular topology of uncountable weight on $ω_1$ there is a non-stationary subset that has uncountable weight as well. Moreover, if all final segments of $ω_1$ have uncountable weight then the assumption of regularity can be dropped. By [6], the analogous statements for the net weight are independent from ZFC. Our proofs of all these results make essential use of elementary submodels. |
| title | Weight, net weight, and elementary submodels |
| topic | Logic General Topology 54A25, 03E05, 03E10, 03E35 |
| url | https://arxiv.org/abs/2503.20061 |