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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.13508 |
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| _version_ | 1866913435301707776 |
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| author | Carlson, Nathan |
| author_facet | Carlson, Nathan |
| contents | Given an open cover $\mathcal{U}$ of a topological space $X$, we introduce the notion of a star network for $\mathcal{U}$. The associated cardinal function $sn(X)$, where $e(X)\leq sn(X)\leq L(X)$, is used to establish new cardinal inequalities involving diagonal degrees. We show $|X|\leq sn(X)^{Δ(X)}$ for a $T_1$ space $X$, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a ccc space with a regular $G_δ$-diagonal has cardinality at most $\mathfrak{c}$, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the property $Ue(X)\leq\min\{aL(X),e(X)\}$ and use the Erdős-Rado theorem to show that $|X|\leq 2^{Ue(X)\overlineΔ(X)}$ for any Urysohn space $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On diagonal degrees and star networks Carlson, Nathan General Topology 54A25, 54D10 Given an open cover $\mathcal{U}$ of a topological space $X$, we introduce the notion of a star network for $\mathcal{U}$. The associated cardinal function $sn(X)$, where $e(X)\leq sn(X)\leq L(X)$, is used to establish new cardinal inequalities involving diagonal degrees. We show $|X|\leq sn(X)^{Δ(X)}$ for a $T_1$ space $X$, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a ccc space with a regular $G_δ$-diagonal has cardinality at most $\mathfrak{c}$, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the property $Ue(X)\leq\min\{aL(X),e(X)\}$ and use the Erdős-Rado theorem to show that $|X|\leq 2^{Ue(X)\overlineΔ(X)}$ for any Urysohn space $X$. |
| title | On diagonal degrees and star networks |
| topic | General Topology 54A25, 54D10 |
| url | https://arxiv.org/abs/2407.13508 |