An isometry is a map which preserves distances between points. Isometries exist in any space in which a distance function is defined, i.e. an arbitrary abstract metric space. In the particular case where we take our space to be the usual Euclidean plane or Euclidean 3-space ( or with the standard Euclidean metric), the isometries are known as rigid motions and two sets which can be transformed onto each other by an isometry are said to be congruent.
Isometries are injective
Since for any metric we have whenever , it follows that every isometry must be an injection.
Suppose otherwise. Then there is some non-injective isometry . Since is not injective, we know such that and . But while , and this contradicts the fact that is an isometry.
Note that this does not mean that isometries are necessarily bijections. Consider, for example, the discrete metric on the integers, such that and . It is simple to verify that this is a metric space. The map given by is an isometry but it is not surjective.
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