TITLE: Examples of non-metrizable spaces 
QUESTION [34 upvotes]: 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.

REPLY [6 votes]: Sorgenfrey line is not metrizable, since it is separable but does not satisfy the second axiom of countability.