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Homemetric equivalence

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# 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.

Defines:

equivalent

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54E35*no label found*

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