论文标题
拓扑空间的主导和固定对
Dominating and pinning down pairs for topological spaces
论文作者
论文摘要
我们称一对无限的红衣主教$（κ，λ）$，$κ>λ$是拓扑空间$ x $的主导（分别固定），如果每个子集$ a $ a $ a $ a $ x $ a $ x $ a $ x $（resp。proces$ \ mathcal {u \ mathcal {u} $ x $ be $ be $ be $ be be be be be be b le be b le be \ be λ$使得$ a \ subset \ overline {b} $（in \ mathcal {u} $中的每个$ u \ for每个$ u \ for \ cap u \ ne \ emptyset $）。显然，统治对也是$ x $的固定对。我们的定义概括了[GTW] resp中引入的概念。 [bt]集中在$（2^λ，λ）$的形式上。 本文的主要目的是回答[gtw]和[bt]的大量问题，该问题询问了$ x $上的某些条件，以及假设$（2^λ，λ）$或$（（（2^λ）^+，λ）$是$ x $ x $ d（x $ d（x $ d（x）$ d（x）$ d（x）$λ。 [BT] A. Bella，V.V。 tkachuk，指数密度与指数统治，预印本 [GTW] G. Gruenhage，V.V。 Tkachuk，R.G。威尔逊（Wilson），由小集与密度，拓扑及其应用的统治282（2020）
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)
