On metric orbit spaces and metric dimension (Q340732)

The metric dimension of a metric space~\((X,d)\) is defined as the minimum cardinality of a subset \(A\) of \(X\) such that the map \(x\mapsto\langle d(x,a):a\in A\rangle\) from~\(X\) to~\(\mathbb{R}^A\) is injective. Simple triangulation shows that, e.g., \(\mathbb{R}^n\)~has metric dimension~\(n+1\). The author proves that if a group \(\Gamma\) of isometries acts properly discontinuously on~\(X\) then the metric dimension of the orbit space~\(X/\Gamma\) is not larger than that of~\(X\) itself. A properly discontinuous action is one such that for each compact set \(K\) the set of~\(g\) for which \(K\cap gK\neq\emptyset\) is finite.