Metric Space

Metric Space

A set of data with information about the distance between each data point. The distances are measured by the metric/distance function.

Graphs are an example of a metric space.

Definition

A metric space is an ordered pair, like $(M,d)$ where $M$ is a set, and each element in the set is a point. $d$ is a metric on the set (a function). $d:M\times{}M\rightarrow{}R$ , where all points satisfy these axioms $x,y,z\in{M}$ : 1) point to self is 0: $d(x,x) = 0$ 2) distance between two points is always positive $if x\neq{y}, \text{then } d(x,y) \gt 0$ 3) x to y and y to x is symmetric $d(x,y)=d(y,x)$ 4) Triangle inequality is true $d(x,z)\leq{d(x,y)+d(y,z)}$ $M$ is often written in place of $(M,d)$