Phase spaces of distal minimal flows (Q5932686)

In the framework of the paper under review, a flow is a triple \((G,X,\pi)\), where \(G\) is a topological group acting continuously on the compact Hausdorff space \(X\), \(\pi:G\times X\to X\) is the action. If there is no proper closed non-empty set \(F\subseteq X\), such that \(\pi^{g}(F)\subseteq F\), \(\forall g\in G\), the flow \((G,X,\pi)\) is called minimal. The flow is distal if for each pair of distinct points \(x,y\in X\) there is an open covering \(\mathcal U\) of the topological space \(X\) such that for any \(g\in G\) and \(U\in \mathcal U\), \(\pi^{g}(x)\notin U\) or \(\pi^{g}(y)\notin U\). In order to state the main results of the paper recall that a topological space \(X\) is a Dugundji space if for each closed subset \(A\) of the Cantor cube \(D^{\kappa}\), and a continuous map \(f:A\to X\) there is a continuous map \(F:D^{\kappa}\to X\), such that \(F_{|A}=f\). A group \(G\) is called \(\omega\)-bounded if for any open neighbourhood \(V\) of the identity element there is a countable set \(A\subseteq G\) such that \(G=AV\). The author proves that the phase space \(X\) of a minimal and distal flow \((G,X,\pi)\), with an arbitrary acting group is a \(\kappa\)-metrizable space. When the acting group is \(\omega\)-bounded, then the phase space \(X\) is a Dugundji space.