definition of 'metric space' isometry
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of 'metric space' isometry.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M_1\): \(\in \{\text{ the metric spaces }\}\), with any metric, \(dist_1\)
\( M_2\): \(\in \{\text{ the metric spaces }\}\), with any metric, \(dist_2\)
\(*f\): \(: M_1 \to M_2\), \(\in \{\text{ the maps }\}\)
//
Conditions:
\(\forall m_1, m_2 \in M_1 (dist_2 (f (m_1), f (m_2)) = dist_1 (m_1, m_2))\)
//
2: Note
Prevalently, just "isometry" is used meaning ''metric space' isometry', but as we use also "isometry between Riemannian manifolds with boundary", "'normed vectors space' isometry", e.t.c., we use "'metric space' isometry".
Often, "isometry" is used requiring bijectivity, but this definition does not require that because otherwise, we would not have any name for the non-bijective case, while we can just use "bijective isometry" for the bijective case.
