Ivan Shishkin, Birch Grove

Metric

Definition / Topology /

Also known as: Distance function

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A metric on a set XX is a function d:X×X→R+d:X\times X \rightarrow \R_{+} such that, for all x,y,z∈Xx,y,z\in X:

  1. Separation: d(x,y)=0⇔x=yd(x,y)=0 \Leftrightarrow x=y
  2. Symmetry: d(x,y)=d(y,x)d(x,y)=d(y,x)
  3. Triangle inequality: d(x,z)≤d(x,y)+d(y,z)d(x,z)\leq d(x,y)+d(y,z)
Remarks
  • The pair (X,d)(X,d) is then called a metric space.
  • d(x,y)d(x,y) is called distance between xx and yy.

Practice this concept with exercises

1 / 3

  • Let XX be a non-empty set. Show that the function d ⁣:X×X→Rd\colon X\times X\to \R such that:
    ∀x,y∈X,d(x,y)={1 if x≠y0 otherwise \forall x,y\in X, d(x, y)= \begin{cases}1 & \text { if } x\neq y \\ 0 & \text { otherwise }\end{cases}is a metric on any set XX. It is called the discrete metric.

    Open exercise
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Which regions are unit disks?
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