Definition of Path: Are the Two Definitions Equivalent?

[JG89]

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?

 

[Hurkyl]

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)

 

[darkside00]

ahhh, are you saying a=0, b=1 for your some a,b?

 

[l'Hôpital]

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).