MathGloss

A metric space is a set XX together with a function ρ:X×X→R\rho: X\times X \to \mathbb R such that for all x,y,z∈Xx,y,z \in X, the following properties are satisfied:

  1. ρ(x,y)=0\rho(x,y) = 0 if and only if x=yx=y.
  2. ρ(x,y)=ρ(y,x)\rho(x,y) = \rho(y,x).
  3. ρ(x,z)≤ρ(x,y)+ρ(y,z)\rho(x,z) \leq \rho(x,y) + \rho(y,z).

Property 3 is called the triangle inequalityand the function ρ\rho is called a metric. The metric generates a topology on XX where the open sets GG are those for which there exists ε>0\varepsilon > 0 for all x∈Gx \in G such that y∈X∣ρ(x,y)<ε⊂G{y\in X\mid \rho(x,y) < \varepsilon}\subset G.

Wikidata ID: Q180953

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