Definition:Isometry (Metric Spaces)

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

Let $$\phi: M_1 \to M_2$$ be a bijection such that $$\forall a, b \in M_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$$.

Then $$\phi$$ is called an isometry.

Isometry Into
When $$\phi: M_1 \to M_2$$ is not actually a surjection, but satisfies the other conditions for being an isometry, then $$\phi$$ can be called an isometry into $$M_2$$.