Equivalence Relation
If is a quasi-isometry, then there exists a quasi-isometry . Indeed, may be defined by letting be any point in the image of that is within distance of, and letting be any point in .
Since the identity map is a quasi-isometry, and the composition of two quasi-isometries is a quasi-isometry, it follows that the relation of being quasi-isometric is an equivalence relation on the class of metric spaces.
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