Definition of Metric Spaces and Distance Functions

Let $S$ be a set. We say that $S$ forms a metric space if there exists a function $d:S\times S\to {\mathbb{R}}_{\ge 0}$, called a metric or distance function, which satisfies the following axioms:

1. Non-Negativity: Given $x,y\in S$, $d(x,y)=0⟺x=y$, i.e. two points are zero distance apart, only if they are the same point.
2. Symmetry: For all $x,y\in S$, $d(x,y)=d(y,x)$, i.e. it doesn’t matter which direction you measure the distance between two points, it should result in the same distance.
3. Triangle Inequality: For all $x,y,z\in S$, $d(x,z)\le d(x,y)+d(y,z)$, i.e. the direct distance between two points is always less than the indirect route.

Informally, a space with the concept of measuring distance.