Definition:Isometry (Metric Spaces)/Into

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

Let $\phi: A_1 \to A_2$ be an injection such that:

$\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$

Then $\phi$ is called an isometry (from $M_1$) into $M_2$.

That is, an isometry (from $M_1$) into $M_2$ is an isometry which is not actually a surjection, but satisfies the other conditions for being an isometry.

Also see

  • Results about isometries can be found here.