Franklin's Notes

Infinity-metric space

An $\infty$-metric space $(M,d)$ is a generalization of a metric space in which the distance function is valued over $\mathbb R^{\geq 0}\cup {\infty}$ rather than $\mathbb R^{\geq 0}$. The triangle inequality is adjusted by defining $x+\infty=\infty$ for any real number $x$, and $\infty+\infty = \infty$. A distance of $\infty$ can be interpreted as indicating that two points are "inaccessible" from each other.

The category $\mathsf{InfMet}$ (whose morphisms are distance-non-increasing functions between $\infty$-metric spaces) is in some ways "nicer" than the traditional category $\mathsf{Met}$. For instance, this category admits coproducts : the coproduct of $(M,d)$ and $(M',d')$ in $\mathsf{InfMet}$ is (up to isomorphism) the metric space $(M^\ast,d^\ast)$ where $M^\ast$ is the disjoint union of $M,M'$ and $d^\ast(x,y)$ is equal to $d(x,y)$ or $d'(x,y)$ if $x,y$ are respectively both contained in $M$ or $M'$ component of the disjoint union, and equal to $\infty$ otherwise.




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