Pseudometric Function

up:: Pseudometric Space

A function $\tilde{d}:M\times M\to {\mathbb{R}}_0^{+}$ is said to be a pseudometric if it satisfies:

1. Self-non-degeneracy: $\forall x\in X,d(x,x)=0$
2. Triangle Inequality: $\forall x,y,z\in X,d(x,z)\le d(x,y)+d(y,z)$
3. Positivity: $\forall x,y\in X,d(x,y)\ge 0$
4. Commutativity: $\forall x,y\in X,d(x,y)=d(y,x)$

It is essentially a Metric Function but with the possibility of different points with distance equal to $0$.

Sufficient conditions for a metric

One can shrink these conditions to two:

1. $\forall x\in X,d(x,x)=0$
2. (Modified) Triangular Inequality: $\forall x,y,z\in X,d(x,z)\le d(x,y)+d(z,y)$

From those two, one can prove the other conditions:

• $1\wedge 2⟹3$: $0=d(x,x)\le d(x,y)+d(x,y)\square$
• $1\wedge 2⟹4$: $d(x,y)\le d(x,x)+d(y,x)$ and $d(y,x)\le d(y,y)+d(y,x)\square$

Note that Every metric function is a pseudometric function, since metrics are more specific (have more properties) than pseudometrics.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 24