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Math Help - isometric in metric space

  1. #1
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    isometric in metric space

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

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

    My question is
    Isometry should be a bijective function between A_{n-1} and R_{n-1}. (a) does not look bijective for me if each x_{k} in A_{n-1} maps to x_{k} in R_{n-1}. ( how do we map 0? it should be bijective by definition of isometry)
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  2. #2
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    And isometry does not need to be a bijection.
    It just needs to preserve distances.
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  3. #3
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    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?
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  4. #4
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    Quote Originally Posted by aliceinwonderland View Post
    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 11,7th Post!
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