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` d_{X}(`a`,`b`) ≤
d_{Y}(f(`a`),f(`b`)) ≤
`C` d_{X}(`a`,`b`)

for all

`a`,

`b`∈

**X**. That is, f

distorts

distances by no more than a

factor of

`C`.

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 **R**^{d} (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.