All pseudometric spaces induce metric spaces

up:: Pseudometric Space

Given a Pseudometric Space , one can create the Equivalence Relation

Given this relation, we can define the Metric Function

for in the Quotient Space.

It is independent on class representatives. Let . Then

Similarly,

Therefore, , and thus is unambiguous, and is a metric function on the quotient space .

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References

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