Distance (metric)

Let $X$ be a set . A metric (or distance function) on $X$ is a function

such that for all $x,y,z\in X$:

• (Positive definiteness) $d(x,y)=0$ iff $x=y$.
• (Symmetry) $d(x,y)=d(y,x)$.
• (Triangle inequality) $d(x,z)\le d(x,y)+d(y,z)$.

Metrics quantify “closeness” abstractly. Most of analysis can be formulated in terms of a metric, including convergence , continuity , compactness , and completeness .

Examples:

• On $\mathbb{R}$, $d(x,y)=|x-y|$ is a metric.
• On $\mathbb{R}^k$, $d(x,y)=\|x-y\|_2$ (Euclidean distance) is a metric.
• On any set $X$, the discrete metric $d(x,y)=0$ if $x=y$ and $1$ otherwise is a metric.