Extended Distance Function

Last updated Nov 1, 2022

Let $S$ be a Set. An Extended Distance Function on $S$ is a Function of the form $d: S^{2} \to \overline{\mathbb{R}_{\geq 0}}$ that satisfies the following properties:

1. $d(x, y) = 0$ If and Only If $x = y$ $\forall x,y \in S$.
2. $d(x, y) = d(y, x)$ $\forall x,y \in S$
3. Triangle Inequality: $d(x, z) \leq d(x, y) + d(y, z)$ $\forall x,y,z \in S$