Isometrisable group actions (Q2814426)

If \(X\) is a metrisable topological space and there exists a compatible invariant distance for an action by homeomorphisms of some group \(G\) on \(X\), then the action \(G \curvearrowright X\) is said to be isometricable.NEWLINENEWLINEThe authors introduce a topological property of \(G \curvearrowright X\) which under certain conditions is equivalent to isometrisability.NEWLINENEWLINEDefinition. \(G \curvearrowright X\) is uniformly topologically equicontinuous if for any \(y \in X\) and any open \(V \ni y\), there exists an open \(U\) with \(y\in U\subseteq V\) such that for all \(x \in X\) there exists an open neighbourhood \(W\) of \(x\) satisfying NEWLINE\[NEWLINE\forall g \in G (gW \cap U \neq 0) \Rightarrow gW \subseteq V.NEWLINE\]NEWLINE The main result of the paper is:NEWLINENEWLINETheorem. Let \(X\) be a second countable metrisable space, and \(G\) be a group acting on \(X\) by homeomorphisms. Then the action \(G \curvearrowright X\) is isometrisable if and only if it is uniformly topologically equicontinuous.