The title is the question, but let me explain. Let $\mathbb{L}$ denote the Sorgenfrey line. I and a friend were trying to develop some of the properties of the sorgenfrey line. (if it's metrizable, paracompact, or whatevs.) And we've stumbled upon the following problem:
Can one define a metric and a group operation in $\mathbb{L}$ that yields the usual order-induced topology and that makes it a topological group? What about semitopological group or something weaker?
So far we've introduced the metric as follows:
$\mathbb{L}$ = $(0,1)$ x $\mathbb{R}$ and x = (t,r), \ y = (t',r') $\ \in$ $\mathbb{L}$
$D(x,y) = 1$ if t \neq t'
$D(x,y) =$ $\frac{\parallel x - y \parallel}{1 + \parallel x - y \parallel}$ if t = t'
We were trying to find countinous group operations unsuccessfully.
Many thanks.