Metric Space

A metric space is a set $X$ paired with a real-valued metric function $d:X\to X\to {\mathbb{R}}^{+}$, such that the following properties hold,

• identity of idiscernibles: $d(x,y)=0$ iff $x=y$ for all $x,y\in X$
• symmetry: $d(x,y)=d(y,x)$ for all $x,y\in X$
• triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$.