The class $C(ω_1)$ and countable net weight
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909119163662336 |
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| author | Juhász, István Soukup, Lajos Szentmiklóssy, Zoltán |
| author_facet | Juhász, István Soukup, Lajos Szentmiklóssy, Zoltán |
| contents | Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class $C(ω_1)$ of topological spaces $X$ having the property that for every neighborhood assignment $\{U(y) : y \in Y\}$ with $Y \in [X]^{ω_1}$ there is $Z \in [Y]^{ω_1}$ such that $$Z \subset \bigcap \{U(z) : z \in Z\}.$$ It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint.
In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C(ω_1)$ has countable net weight, answering a question that was raised by Hart and Kunen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11014 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The class $C(ω_1)$ and countable net weight Juhász, István Soukup, Lajos Szentmiklóssy, Zoltán General Topology 54A25, 54A35, 54D99, 03E35 Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class $C(ω_1)$ of topological spaces $X$ having the property that for every neighborhood assignment $\{U(y) : y \in Y\}$ with $Y \in [X]^{ω_1}$ there is $Z \in [Y]^{ω_1}$ such that $$Z \subset \bigcap \{U(z) : z \in Z\}.$$ It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C(ω_1)$ has countable net weight, answering a question that was raised by Hart and Kunen. |
| title | The class $C(ω_1)$ and countable net weight |
| topic | General Topology 54A25, 54A35, 54D99, 03E35 |
| url | https://arxiv.org/abs/2307.11014 |