1. There is no distance between an element and itself: $d(x_i,x_i)=0$.
2. If the distance between two elements is $0$, those elements are equivalent: $d(x_i,x_j)=0\implies x_i=x_j$.
3. All distances are non-negative: $d(x_i,x_j)\ge0$.
4. The distance between two elements is the same in either direction: $d(x_i,x_j)=d(x_j,x_i)$.
5. The distance between two elements is less than or equal to the sum of the distances between those elements and a third: $d(x_i,x_j)\le d(x_i,x_k)+d(x_j,x_k)$