I think it's cute to introduce talking about topologies on function
spaces by suggesting that one can -choose- different topologies on the
real numbers!

In addition to the standard topology, you can also take the trivial
topology (every sequence converges to everything!),
or the discrete topology (only convergent sequences are eventually
constant).

(note that the discrete topology is metrizable, while the trivial
topology isn't even Hausdorff!)

(the zariski topology isn't terribly understandable from this point of
view, as it's ``just another non-Hausdorff topology'' from the point
of view of sequences)

(note that the statement:
$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dotsb$
may -approach- $2$, but they're never actually -equal-
makes sense in the -discrete- topology!
limits -mean- take convergence at infinity
