# metric equivalence

Let $X$ be a set equipped with two metrics $\rho$ and $\sigma$. We say that $\rho$ is equivalent to $\sigma$ (on $X$) if the identity map on $X$, is a homeomorphism between the metric topology on $X$ induced by $\rho$ and the metric topology on $X$ induced by $\sigma$.

For example, if $(X,\rho)$ is a metric space, then the function $\sigma:X\to\mathbb{R}$ defined by

 $\sigma(x,y):=\frac{\rho(x,y)}{1+\rho(x,y)}$

is a metric on $X$ that is equivalent to $\rho$. This shows that every metric is equivalent to a bounded metric.

Title metric equivalence MetricEquivalence 2013-03-22 19:23:11 2013-03-22 19:23:11 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 54E35 equivalent