Definition:Distance-Preserving Mapping

Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces, pseudometric spaces, or quasimetric spaces.

Let $\phi: M_1 \to M_2$ be a mapping such that:

$\forall a, b \in M_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

Then $\phi$ is called a distance-preserving mapping.

Also see

• Definition:Isometry (Metric Spaces)

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces