群$S^3$和$SO(3)$没有不变的二元$k$ -网络
The groups $S^3$ and $SO(3)$ have no invariant binary $k$-network
摘要：
拓扑空间$X$的闭子集族$\mathcal N$称为{ \em闭$k$ -网}，如果对于每个开集$U\subset X$和紧子集$K\subset U$，存在一个有限的子集$\mathcal F\subset\mathcal N$，其中有$K\subset\bigcup\F\subset \mathcal N$ .一个紧空间$X$称为{ \em超紧}，如果它允许一个闭的$k$ -网络$\mathcal N$，它是{ \em二进制}，在这个意义上，每个链接的子家族$\mathcal L\subset\mathcal N$是中心的。拓扑群$G$中的闭$k$网络$\mathcal N$是{ \em不变量}，如果对于每个$A\in\mathcal N$和$x,y\in G$都有$xAy\in\mathcal N$.根据Kubietan和Turek的结果，每个紧（交换）拓扑群都有一个（不变）二元闭$k$ -网.本文证明了紧拓扑群$S^3$和$\SO(3)$不存在不变二元闭$k$ -网。
英文摘要：
A family $\mathcal N$ of closed subsets of a topological space $X$ is called a {\em closed $k$-network} if for each open set $U\subset X$ and a compact subset $K\subset U$ there is a finite subfamily $\mathcal F\subset\mathcal N$ with $K\subset\bigcup\F\subset \mathcal N$. A compact space $X$ is called {\em supercompact} if it admits a closed $k$-network $\mathcal N$ which is {\em binary} in the sense that each linked subfamily $\mathcal L\subset\mathcal N$ is centered. A closed $k$-network $\mathcal N$ in a topological group $G$ is {\em invariant} if $xAy\in\mathcal N$ for each $A\in\mathcal N$ and $x,y\in G$. According to a result of Kubiś and Turek, each compact (abelian) topological group admits an (invariant) binary closed $k$-network. In this paper we prove that the compact topological groups $S^3$ and $\SO(3)$ admit no invariant binary closed $k$-network.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
