A function f:X→Y between two metric spaces is called a quasi-isometry (or a near isometry) if there exists a constant C for which
1/C dX(a,b) ≤
for all a
. That is, f distort
s by no more than a factor
A quasi-isometry is always one-to-one, but needn't be onto.
Isometry between metric spaces is a very strong condition indeed. Often, a weaker condition is useful. For instance, any two norms on Rd (d<∞) induce quasi-isometric metric spaces; except when the two norms are multiples of each other, they never induce isometry.
If f is a quasi-isometry which is also onto, then the topologies of the two metric spaces are equivalent: convergence in the one implies convergence in the other.