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Definition of path

  1. May 26, 2010 #1
    My real analysis book says that a path from two points p, q in a metric space M is a continuous function f: [a,b] --> M such that f(a) = p and f(b) = q, for some a and some b. But when I read other definitions, it says a path from two points p, q in a metric space M is a continuous function f: [0,1] ---> M such that f(0) = p, f(1) = q.

    Are the two definitions equivalent?
     
  2. jcsd
  3. May 26, 2010 #2

    Hurkyl

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    Yes and no.

    In the most literal sense, they are clearly inequivalent -- there exist paths by the first definition that obviously aren't paths by the second definition. (just choose (a,b) to be anything but (0,1))


    However, I can't think of any application of paths that really cares about the difference between the two definitions.


    (note that you can always compose with your favorite order isomorphism [0,1]-->[a,b] to convert any path [a,b]-->M into a path [0,1]-->M)
     
  4. May 26, 2010 #3
    ahhh, are you saying a=0, b=1 for your some a,b?
     
  5. May 27, 2010 #4
    if f : [a,b] --> M , g : [0,1] ---> M. and you want f(a) = g(0), and f(b) = g(1).

    let g(x) = f(a + (b-a)x).
     
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