Non-Metrizable Topological Spaces

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.

Any topological space which is not Hausdorff is non-metrizable.

Maybe you would like the Zariski topology and things like that.