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:


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


Your help is appreciated.



Examples of non-metrizable spaces 
I'm not an expert, but I believe the space of Distributions (in any number of variables), (a.k.a. generalised functions, including the Dirac delta, its derivatives, etc.) as used in PDE theory, is a topological vector space, but *non-metrisable*; even though sequences *are* sufficient to do everything.


However, the subspace of *tempered* (Schwartz) distributions, as used in Fourier Analysis, *is* metrisable; it is a Fréchet space.


