# Thread: isometric in metric space

1. ## isometric in metric space

My textbook says that
For n>=2, consider the set
$\displaystyle A_{n-1} = \{ x = (x_{1},....,x_{n}) \in R^{n} : x_{n}=0\}$ with metric induced by d (usual metric). Then $\displaystyle A_{n-1}$ is a subset of $\displaystyle R_{n}$ and is isometric to $\displaystyle R_{n-1}$ under correspondence

(a) $\displaystyle (x_{1},...,x_{n-1}, 0) \leftrightarrow (x_{1},...,x_{n-1})$

My question is
Isometry should be a bijective function between $\displaystyle A_{n-1}$ and $\displaystyle R_{n-1}$. (a) does not look bijective for me if each $\displaystyle x_{k}$ in $\displaystyle A_{n-1}$ maps to $\displaystyle x_{k}$ in $\displaystyle R_{n-1}$. ( how do we map 0? it should be bijective by definition of isometry)

2. And isometry does not need to be a bijection.
It just needs to preserve distances. 3. According to mathworld, an isometry is defined as follows:

"A bijective map between two metric spaces that preserve distances, i.e,
d(f(x), f(y)) = f(x, y),
where f is the map and d(a,b) is the distance function"

Am I confusing something?

4. Originally Posted by aliceinwonderland According to mathworld, an isometry is defined as follows:

"A bijective map between two metric spaces that preserve distances, i.e,
d(f(x), f(y)) = f(x, y),
where f is the map and d(a,b) is the distance function"

Am I confusing something?
Not according to this. This is Mine 11,7  th Post!

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