Every metric function is a pseudometric function

up:: Pseudometric Function

Note that, by definition of a Metric Function, it satisfies more conditions than a Pseudometric Function; thus, it is “more specific” than a pseudometric.

Therefore, every metric function is trivially a pseudometric function ─ with the extra condition that $d(x,y)=0⟹x=y$.

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References

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