Q:

Topology that induced by uniform structure

Let $(X,d)$ a metric space .Calculate the topology induced by uniformity that induced by pseudo metric d ( $U_{d} $)? Is topology induced by metric ?
where $U_d=\{v\subseteq X\times X : v_\epsilon \subseteq v\} $ and $ v_\epsilon =\{(x,y)\in X \times X: d(x,y)< \epsilon \} $

A:

Of course. $$B_d(x,r)= v_r[x]$$ so both topologies (the uniform one and the metric one) have the same local bases.

