拓扑空间的控制对和牵制对
Dominating and pinning down pairs for topological spaces
摘要：
我们称一对无限基数$(κ,λ)$，其中$κ> λ$是支配的（分别是）。对于拓扑空间$X$，如果对于$X$的每个子集$A$（分别为$X$中的非空开集的家族$\mathcal{U}$）的基数$\le κ$存在基数$\le λ$的$B \subset X$使得$A \subset \overline{B}$（分别每个$U \in \mathcal{U}$的$B \cap U \ne \emptyset$）。显然，对于$X$，支配对也是牵制对。我们的定义推广了[GTW]中分别引入的概念。[BT]其集中于$(2^λ,λ)$形式的对。本文的主要目的是回答大量的众多问题，从[GTW]和[BT]，询问如果某些条件下的空间$X$连同假设$(2^λ,λ)$或$((2^λ)^+,λ)$是一个牵制对或\dominating对$X$将意味着$d(X) \le λ$。[BT]A. Bella，V.V. Tkachuk，指数密度与指数支配，预印本[GTW] G。Gruenhage，V.V. Tkachuk，R.G. Wilson，Domination by small sets versus density，Topology and its Applications 282（2020）
英文摘要：
We call a pair of infinite cardinals $(κ,λ)$ with $κ> λ$ a dominating (resp. pinning down) pair for a topological space $X$ if for every subset $A$ of $X$ (resp. family $\mathcal{U}$ of non-empty open sets in $X$) of cardinality $\le κ$ there is $B \subset X$ of cardinality $\le λ$ such that $A \subset \overline{B}$ (resp. $B \cap U \ne \emptyset$ for each $U \in \mathcal{U}$). Clearly, a dominating pair is also a pinning down pair for $X$. Our definitions generalize the concepts introduced in [GTW] resp. [BT] which focused on pairs of the form $(2^λ,λ)$. The main aim of this paper is to answer a large number of the numerous problems from [GTW] and [BT] that asked if certain conditions on a space $X$ together with the assumption that $(2^λ,λ)$ or $((2^λ)^+,λ)$ is a pinning down pair or \dominating pair for $X$ would imply $d(X) \le λ$. [BT] A. Bella, V.V. Tkachuk, Exponential density vs exponential domination, preprint [GTW] G. Gruenhage, V.V. Tkachuk, R.G. Wilson, Domination by small sets versus density, Topology and its Applications 282 (2020)
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