Functions between metric spaces
Functions between metric spaces [ edit ] Euler diagram of types of functions between metric spaces. Unlike in the case of topological spaces or algebraic structures such as groups or rings , there is no single "right" type of structure-preserving function between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that ( � 1 , � 1 ) and ( � 2 , � 2 ) are two metric spaces. The words "function" and "map" are used interchangeably. Isometries [ edit ] Main article: Isometry One interpretation of a "structure-preserving" map is one that fully preserves the distance function: A function � : � 1 → � 2 is distance-preserving [12] if for every pair of points x and y in M 1 , � 2 ( � ( � ) , � ( � ) ) = � 1 ( � , � ) . It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is calle