diff options
Diffstat (limited to 'pages/general-topology')
| -rw-r--r-- | pages/general-topology/metric-spaces/index.md | 10 |
1 files changed, 9 insertions, 1 deletions
diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md index 52b2b4c..6aef400 100644 --- a/pages/general-topology/metric-spaces/index.md +++ b/pages/general-topology/metric-spaces/index.md @@ -60,8 +60,16 @@ Let $(X,d)$ be a (semi-)metric space. The proofs are straightforward. +{% definition Isometry %} +Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces. +We say that a mapping $f : X \to Y$ is *isometric* or an *isometry* if it obeys +$d_Y \big\lparen f(x),f(x') \big\rparen = d_X(x,x')$ for all $x,x' \in X$. +{% enddefinition %} + +As a consequence of **(M1)**, +every isometry is injective. + TODO -- isometry - metric induced by a norm - metric product |