Topological spaces vs. metric spaces Are there any "realistic" examples of topological spaces that are not metric spaces. You are free to invent your own definition of "realistic". But, at a minimum, a realistic example is one that occurs naturally as part of some other problem, not something that is fabricated to provide a counterexample or a topology homework exercise. One good example is the topological vector space of all functions $f:\mathbb{R} \to \mathbb{R}$ under pointwise convergence. Any others?

If such examples are difficult to find, maybe there are results in the opposite direction: are there results that say something like "every topology that satisfies conditions X, Y, Z can be generated from a metric"? One example: a compact Hausdorff space is metrizable if and only if it is second-countable. Any other theorems like that?

Edit
I found this page, which does a pretty good job of answering my second question. So, that just leaves the first one.
Ordinal spaces occur naturally, and any uncountable ordinal space is not metrizable.

You can also talk about Moore spaces, Stone-Cech compactification of $\Bbb N$ (which is compact but has is too big to be metrizable), there are Zariski topologies which are often non-metrizable, and there are plenty of Cantor cubes which are naturally occurring in set theory.
The projection of the complex plane onto the real axis gives a simple, naturally occurring, example of a topological space that is not a metric space.  Define d(z1, z2) = |Re(z1)-Re(z2)|.  This is not a metric on the complex plane because d(z1, z2) = 0 does not imply z1=z2.  But the open sets defined the obvious way does define a topology on C.  Any non-trivial projection onto a metric space can give a similar example.