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 $\sigma :X\to ℝ$ 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.