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Math Help - Isometric imbedding

  1. #1
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    Isometric imbedding

    Let X,Y be metric spaces with metrics d1,d2 respectively.
    let f:X-->Y be a function with the property that for all pair of points x1,x2 of X, d2(f(x1),f(x2))=d1(x1,x2).
    Show that f is an imbedding..

    thnak you in advance
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  2. #2
    Senior Member Sampras's Avatar
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    Quote Originally Posted by math.dj View Post
    Let X,Y be metric spaces with metrics d1,d2 respectively.
    let f:X-->Y be a function with the property that for all pair of points x1,x2 of X, d2(f(x1),f(x2))=d1(x1,x2).
    Show that f is an imbedding..

    thnak you in advance

    Take L = C = 1.

    So  Ld_{1}(x,y) \leq d_{2}(f(x_1), f(x_2)) \leq Cd_{1}(x,y) which is definition of embedding.
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  3. #3
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    thank you for ur response..but we have not yet come across this definition of embedding..
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  4. #4
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    i need a little help..i have shown that f is injective and continuous.now i need to show that g:X-->f(x) is a homeomorphism..how do i show g and g^-1(that is g inverse) are continuous..
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