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 $X$ be a set and let $d:X\times X\to\mathbb{R}$ be a map.

(i) $d$ is a metric on $X$ if for all $x,y,z\in X$ the conditions nonnegative, zero, symmetric and triangle below hold.

(ii) $0\le d(x,y)$.

(iii) $d(x,y)=0$ if and only if $x=y$.

(iv) $d(x,y)=d(y,x)$.

(v) $d(x,z)\le d(x,y)+d(y,z)$.

(vi) If $d$ is a metric on $X$, the pair $(X,d)$ is a metric space, and the elements of $X$ are its points.