A quasi-isometry is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example, and its subspace are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in geometric group theory: the Švarc–Milnor lemma states that all spaces on which a group acts geometrically are quasi-isometric.
A quasi-isometry is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example, and its subspace are quasi-isometric, even though one is c.onnected and the other is discrete. The. equivalence relation of quasi-?isometry is important in geometric group theory: the Švarc–Milnor lemma states that al?l spaces on which a group acts geometrically are quasi-isometric.