Pseudometric Function

up:: Pseudometric Space

A function is said to be a pseudometric if it satisfies:

1. Self-non-degeneracy:
2. Triangle Inequality:
3. Positivity:
4. Commutativity:

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

Sufficient conditions for a metric

One can shrink these conditions to two:

2. (Modified) Triangular Inequality:

From those two, one can prove the other conditions:

• :
• : and

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