uniform structure of a metric space

Let $(X,d)$ be a metric space. There is a natural uniform structure on $X$ , which induces the same topology as the metric. We define a subset $V$ of the Cartesian product $X\times X$ to be an entourage if and only if it contains a subset of the form

$V_{\varepsilon}=\{(x,y)\in X\times X:d(x,y)<\varepsilon\}$

for some $\varepsilon$ .

Title	uniform structure of a metric space
Canonical name	UniformStructureOfAMetricSpace
Date of creation	2013-03-22 12:47:18
Last modified on	2013-03-22 12:47:18
Owner	n3o (216)
Last modified by	n3o (216)
Numerical id	6
Author	n3o (216)
Entry type	Derivation
Classification	msc 54E15