Category:Isometries (Metric Spaces)

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This category contains results about isometries in the context of metric spaces.
Definitions specific to this category can be found in Definitions/Isometries (Metric Spaces).

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

Let $\phi: A_1 \to A_2$ be a bijection such that:

$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

Then $\phi$ is called an isometry.

That is, an isometry is a distance-preserving bijection.