All pseudometric spaces induce metric spaces

up:: Pseudometric Space

Given a Pseudometric Space $(M,\tilde{d})$, one can create the Equivalence Relation

$$x\sim y⟺\tilde{d}(x,y)=0$$

Given this relation, we can define the Metric Function

$$d([x],[y]):=\tilde{d}(x,y)$$

for $[x],[y]\in M/\sim$ in the Quotient Space.

It is independent on class representatives. Let $x_1,x_2\in [x],y_1,y_2\in [y]$. Then

$$\tilde{d}(x_1,y_1)\le \tilde{d}(x_1,x_2)+\tilde{d}(x_2,y_2)+\tilde{d}(y_2,y_1)=\tilde{d}(x_2,y_2)$$

Similarly,

$$\tilde{d}(x_2,y_2)\le \tilde{d}(x_2,x_1)+\tilde{d}(x_1,y_1)+\tilde{d}(y_1,y_2)=\tilde{d}(x_1,y_1)$$

Therefore, $\tilde{d}(x_1,y_1)=\tilde{d}(x_2,y_2)$, and thus $d([x],[y])$ is unambiguous, and $d$ is a metric function on the quotient space $M/\sim$.

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References

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