Education: Metric space: Isometry

Metric Space: Isometry

An isometry is a surjective mapping

f: X \rightarrow Y

, where

(X , \delta )

and

(Y , \rho )

are metric spaces and for all

a, b \in X

\delta (a, b) = \rho (f(a), f(b))

In this case,

(X, \delta )

and

(Y, \rho )

are said to be isometric.

Note that the injectivity of

f

follows from the property of preserving distance:

f(a)=f(b)

\implies\rho(f(a),f(b))=0

\implies\delta(a,b)=0

\implies a=b

So an isometry is necessarily bijective.