Consider the Reals as a totally ordered set via its natural order ( linear continuum ).
Such order induces an order topology ( basis is the collection of open-intervals of that order), which ends up being a metrizable space.
The metrizable space has an equivalence-class of metrics ( equivalent metrics ) which generates such order topology, the most famous of which is the euclidean metric $d(x,y) = |x-y|$.
The mentioned metric is induced by the euclidean norm on the 1-dimensional vector space of the reals.
The mentioned metric is also based on the absolute-value function natural to the unique complete ordered field of the reals ( in every ordered field, we can naturally define the notion of an absolute value function that is enough , with the provided arithmetic, to define a natural metric ).
What i'm most curious about is that it seems that in this case the natural order alone on the set of reals is indirectly inducing a vector space whose natural norm defines a metric that induces the order topology induced by that order.
At the same time, it seems that the order alone on the set of reals is indirectly inducing an ordered field whose natural absolute value function defines a metric that induces the order topology induced by that order on that set.
I'm just finding a bit curious and interesting the relation between a simple order on a set and many possible natural inductions ( vector space whose natural norm defines a metric that induces the order topology or ordered field whose absolute value function defines a natural metric that induces the order topology ) .
Is it only happening because we ordered set into consideration is the reals ( which is pretty important, even required for the notion of metric ) or is this notion more general ?
Any insight about the interplay between order and metric in that particular case ( or generally ) is appreciated.
Thanks in advance.