Open Subset of a Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space, and let $U\subseteq X$.

(i) $U$ is open in $(X,d)$ if for every $x\in U$ there is a real number $r>0$ such that the open ball $B_d(x,r)$ is a subset of $U$.