isometry (original) (raw)

Every isometric mapping is injective, for if x,y∈X1 with x≠y then d1⁢(x,y)>0, and so d2⁢(f⁢(x),f⁢(y))>0, and then f⁢(x)≠f⁢(y). One can also easily show that every isometric mapping is continuous.

An isometric mapping that is surjective (and therefore bijective) is called an isometry. (Readers are warned, however, that some authors do not require isometries to be surjective; that is, they use the term isometryfor what we have called an isometric mapping.) Every isometry is a homeomorphism.

If there is an isometry between the metric spaces (X1,d1) and (X2,d2), then they are said to be isometric. Isometric spaces are essentially identical as metric spaces, and in particular they are homeomorphic.

Given any metric space (X,d), the set of all isometries X→X forms a group under composition. This group is called the isometry group(or group of isometries) of X, and may be denoted by Iso⁡(X) or Isom⁡(X). In general, an (as opposed to the) isometry group (or group of isometries) of X is any subgroup of Iso⁡(X).