I've started studying topology, and the impression I get is that it's all about studying spaces without using a metric. So we have to talk about these open sets instead. So basically a topological space is a generalization of a metric space.
What I haven't been able to find, though, is any example of a space I would be interested in that doesn't have a metric! I'm not looking for general "applications of topology", I'm looking for a specific non-metric topological space that is interesting outside of topology, that doesn't have a metric. Preferably as simple as possible; i.e. ideally you should be able to just write it as a set comprehension. But any interesting spaces without metrics qualify.
There are a lot. A few simple examples you can look up are the Sorgenfrey line on $\Bbb R$, the cofinite topology on any infinite set, and for a really easy example, the indiscrete topology on any set with more than one point.
How easy it is to prove these aren't metrizable depends on how much you've seen already, but for the latter two showing that the spaces aren't Hausdorff is probably the easiest way.