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 UniformStructureOfAMetricSpace 2013-03-22 12:47:18 2013-03-22 12:47:18 n3o (216) n3o (216) 6 n3o (216) Derivation msc 54E15