# 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.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Isometries (Metric Spaces)"

The following 15 pages are in this category, out of 15 total.

### D

### E

### I

- Isometric Image of Cauchy Sequence is Cauchy Sequence
- Isometric Metric Spaces are Homeomorphic
- Isometry between Metric Spaces is Continuous
- Isometry is Homeomorphism of Induced Topologies
- Isometry of Metric Spaces is Equivalence Relation
- Isometry of Metric Spaces is Homeomorphism
- Isometry Preserves Sequence Convergence