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

TL;DR

This paper proves that the topological groups $S^3$ and $SO(3)$ do not possess invariant binary closed $k$-networks, contrasting with known results for other compact (abelian) groups.

Contribution

It demonstrates the non-existence of invariant binary closed $k$-networks in specific non-abelian compact groups $S^3$ and $SO(3)$, extending the understanding of topological group structures.

Findings

$S^3$ admits no invariant binary closed $k$-network.

$SO(3)$ admits no invariant binary closed $k$-network.

Contrasts with known results for abelian groups.

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 Kubi\'s 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.