Open Ball in 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, let $x\in X$, and let $r$ be a real number with $r>0$.

(i) The open ball in $X$ with centre $x$ and radius $r$ is the subset

$$B_d(x,r)=\{y\in X:d(x,y)<r\}$$

of $X$.

(ii) For every $y$, $y\in B_d(x,r)$ if and only if $y\in X$ and $d(x,y)<r$.