The groups $S^3$ and $SO(3)$ have no invariant binary $k$-network

Abstract

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.

arXiv comment5 pages

References in corpus (1)

• A decomposition theorem for compact groups with application to supercompactness