Editing Metric space (section)

===Notions of metric space equivalence===
{{See also|Equivalence of metrics}}
Given two metric spaces <math>(M_1, d_1)</math> and <math>(M_2, d_2)</math>:
*They are called '''homeomorphic''' (topologically isomorphic) if there is a [[homeomorphism]] between them (i.e., a continuous [[bijection]] with a continuous inverse).  If <math>M_1=M_2</math> and the identity map is a homeomorphism, then <math>d_1</math> and <math>d_2</math> are said to be '''topologically equivalent'''.
*They are called '''uniformic''' (uniformly isomorphic) if there is a [[uniform isomorphism]] between them (i.e., a uniformly continuous bijection with a uniformly continuous inverse).
*They are called '''bilipschitz homeomorphic''' if there is a bilipschitz bijection between them (i.e., a Lipschitz bijection with a Lipschitz inverse).
*They are called '''isometric''' if there is a (bijective) [[isometry]] between them. In this case, the two metric spaces are essentially identical.
*They are called '''quasi-isometric''' if there is a [[quasi-isometry]] between them.