# Definition:Isometry (Metric Spaces)/Into

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## Definition

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

Let $\phi: A_1 \to A_2$ be an injection 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 (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.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Definition $2.4.9$