Ivan Shishkin, Birch Grove

Metric

Definition / Topology / Stub

Also known as: Distance function

English
EnglishFrançais
This article is a stub
Stub. This concept is still a minimal draft.

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
Problems using this concept (0)

No listed problems link to this concept yet.

Problems using this concept (spoiler) (0)

No listed problems use this concept as a spoiler yet.