What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. Also, it would be cool and informative if you could list some basic topological properties that each of these spaces have. Thanks.
1) Co- countable topology 2) Co- finite topology 3) Sierpiński spaces are example of non metrizable topological spaces.
Normally topology just comes from the basis[generator of a topology], but metric spaces come from the distance function $d$. Then we observe that the open balls gives us the basis & the topology generated from that is the metrizable topology.
Again see all the topological properties concerning open sets such as arbit union of open sets are open, convergence of sequence, continuity[with taking distance in the role] etc hold for both spaces metrizable & non metrizable topological spaces , where the difference lies is concerning distance such as Hausdorff property, $T_1,T_3 $ properties etc.
There are lots of nonmetrizable spaces that are relevant. Not just pathological examples, like the trivial topology.
Wikipedia has a good example: all functions $f:R\rightarrow R$ with the topology of pointwise convergence. Read about it here: What is the Topology of point-wise convergence?.