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.

Title	metric equivalence
Canonical name	MetricEquivalence
Date of creation	2013-03-22 19:23:11
Last modified on	2013-03-22 19:23:11
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54E35
Defines	equivalent