I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other areas of mathematics like analysis or algebra. I know most spaces arising naturally in other areas of mathematics are metrizable because of the Urysohn metrization theorem. But still there must be some examples of non-metrizable spaces.So far I know the following examples:

- Zariski topology
- Weak* topology on $X^{*}$ if X is an infinite dimensional Banach space
- The topological vector space of all functions $f:\mathbb{R}\rightarrow\mathbb{R}\ \ $ under pointwise convergence.

Your help is appreciated.

Counterexamples in Topology? $\endgroup$ismetrizable as long as $X$ is separable $\endgroup$