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Quasimetric on the Sorgenfrey Line

  1. Oct 3, 2007 #1
    1. The problem statement, all variables and given/known data

    Define a quasimetric on the Sorgenfrey Line.


    2. Relevant equations

    I know how to show the distance function is always nonnegative, equal to zero if evaluating the distance of a point from itself, and the triangle inequality. I'm having trouble coming up with the function.


    3. The attempt at a solution

    I have defined d(p,q) = q - p (if q > p) and 0 otherwise. I want to show that the sphere centered at p of radius e is the same as the interval [p, p+e). This will complete the proof.

    Suppose x is in the sphere centered at p of radius e. Then either d(p,x) = 0 or d(p,x) < e
    (IFF) p <= x or x - p < e (IFF) p <= x or x < p + e.

    What bothers me here is the "or". It feels like I need "and" to justify x in [p, p+e). Help?
     
  2. jcsd
  3. Oct 4, 2007 #2

    morphism

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    The function you're proposing doesn't really satisfy the definition of a quasimetric. For example, d(3,1)=0. How about we define d(x,y) = {y-x, if y>=x; 1 otherwise} instead? I think this is a valid quasimetric, although some case-checking is needed to establish the triangle inequality (which I haven't really done in a satisfactory way!).

    Now, let's say we're give the basic open set [x,y) in the Sorgenfrey topology. Consider the cases when |y-x|>=1 and |y-x|<1, and show that we can get a ball to sit inside [x,y) in each case.

    The other direction, that the Sorgenfrey line is finer than our quasimetric topology, is immediate.
     
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