The groups \(S^3\) and \(\mathrm{SO}(3)\) have no invariant binary \(k\)-network (Q2850627)

Let \(\mathcal{A}\) be a family of subsets of a set \(X\). Then \(\mathcal{A}\) is called linked if \(A\cap B\neq \emptyset\) for all \(A, B\in \mathcal{A}\), \(\mathcal{A}\) is called centered if \(\bigcap\mathcal{F}\neq \emptyset\) for any finite subfamily \(\mathcal{F}\) of \(\mathcal{A}\) and \(\mathcal{A}\) is called binary if each linked subfamily of \(\mathcal{A}\) is centered. A family \(\mathcal{N}\) of closed subsets of a topological space \(X\) is called a closed \(k\)-network if for each open subset \(U\) of \(X\) and a compact subset \(K\) of \(U\) there is a finite subfamily \(\mathcal{F}\) of \(\mathcal{N}\) with \(K\subseteq \bigcup\mathcal F\subseteq U\). A compact space \(X\) is called supercompact if it admits a closed \(k\)-network \(\mathcal{N}\) which is binary in the sense that each linked subfamily \(\mathcal{L}\) of \(\mathcal{N}\) is centered. A closed \(k\)-network \(\mathcal{N}\) in a topological group \(G\) is invariant if \(xAy\in \mathcal{N}\) for each \(A\in \mathcal{N}\) and \(x, y \in G\). According to a result of \textit{W. Kubiš} and the second author [Cent. Eur. J. Math. 9, No. 3, 593--602 (2011; Zbl 1235.22007)], which says that each compact (abelian) topological group admits an (invariant) binary closed \(k\)-network. The authors of this paper prove that the compact topological groups \(S^3\) and \(\mathrm{SO}(3)\) admit no invariant binary closed \(k\)-network.