Order vs. metric

One -must- use the order topology (rather than the metric) to define
limits at infinity and functions going off to infinity (essentially);
thus even if one is committed to the metric point of view, one needs
to use the ordering at some point.

One can be more or less committed to the ordering point of view:
if you write stuff like $A - \epsilon < x < A + \epsilon$, then you're
just making a symbolic (literally ;-) change; to be really committed
to the order topology means using $A < x < B$ as the standard open
set.
(contrast Leonard Gillman's Calculus book with his later articles)

Note that there are two natural ways to compactify $\bR$: add in $\pm
\infty$, or add in a single unsigned $\infty$.

"metric = distance"
Metric is a technical word for distance function
(joke about metric system:
 to get within 1 km of f(x), have to be within 1 nm of x ;-)

--------------------------------------------
notes about nets being the natural object in more general topology;
see how they naturally arise in partitions, for instance.
(nets are very natural, and give more insight into sequences:
 in a sequence, the sequence of points stands in for the sequence of
 $1/n$ basic open sets; hence sequences generalize to being enough for
 uniform spaces, in particular metric spaces.
 Since in general the system of open sets is a directed system, nets
 are the natural generalization.)
