关于对角线度和星形网络

Nathan Carlson
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引用次数: 0

摘要

给定拓扑空间 $X$ 的开盖 $\mathcal{U}$,我们为 $\mathcal{U}$ 引入星形网络的概念。相关的心函数 $sn(X)$,其中 $e(X)\leq sn(X)\leq L(X)$,被用来建立涉及对角度的新心不等式。我们证明了 $T_1$ 空间 $X$ 的 $|X|\leqsn(X)^{\Delta(X)}$,从而部分回答了安杰洛-贝拉(Angelo Bella)一直以来的问题。使用 $sn(X)$ 的变化给出了许多进一步的结果。其中一个结果的推论是布扎科娃(Buzyakova)的定理,即具有规则 $G_\delta$ 对角线的 accc 空间最多具有 cardinality $/mathfrak{c}$,以及哥切夫(Gotchev)的三个结果。进一步的结果包括对巴西尔、贝拉和里德博斯定理的逻辑改进,对同一作者一个问题的部分解答,以及哥切夫、特卡琴科和特卡丘克的一个定理。最后,我们定义了具有$Ue(X)\leq\min\{aL(X),e(X)\}$性质的乌里索恩程度$Ue(X)$,并使用埃尔德{H{o}斯-拉多定理证明了对于任意乌里索恩空间$X$,$|X|\leq 2^{Ue(X)\overline\{Delta}(X)}$ 。
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On diagonal degrees and star networks
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 $|X|\leq sn(X)^{\Delta(X)}$ 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 $G_\delta$-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 Erd\H{o}s-Rado theorem to show that $|X|\leq 2^{Ue(X)\overline{\Delta}(X)}$ for any Urysohn space $X$.
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