On maximal groups of isometries

Abstract:

The purpose of this note is to introduce the concept of “Optimal Metrization” for metrizable topological spaces. Let $X$ be such a space, $\rho$ a metric on $X$ and $K(\rho )$ the group of all those homeomorphisms of $X$ onto itself which preserve $\rho$. The metric $\rho$ is said to be “optimal” provided there is no ${\rho ^ \ast }$ with $K({\rho ^ \ast })$ properly containing $K(\rho )$. A space having at least one optimal metric is called “optimally metrizable.” Examples of spaces which are and which are not optimally metrizable are given; it is shown that the real line $R$ is, and that the usual metric is optimal.