The metric dimension of metric spaces (Q380725)

Let \((X,d)\) be a metric space. A non-empty subset \(A\) of \(X\) resolves \((X,d)\) if \(d(x,a)=d(y,a)\) for all \(a\) in \(A\) implies \(x=y\), and if that is so we may regard the distances \(d(x,a)\), where \(a\in A\), as the coordinates of \(x\) with respect to \(A\). The metric dimension of \((X,d)\) is the smallest integer \(k\) such that there is a set \(A\) of cardinality \(k\) that resolves \(X\). The authors derive many properties of the metric dimension. In particular they discus the metric dimension for graphs, the Euclidean space \(\mathbb{R}^n\), the hyperbolic space \(\mathbb{H}^n\), the spherical space \(\mathbb{S}^n\) and for certain subsets of these spaces. In the last section they mention a number of open questions.