@PeterTamaroff Yes, some examples are $\Bbb R$ with basis open sets $[a, b)$, the long line, and $\Bbb R$ with basis open sets of the form $U - \text{countable set}$, where $U$ is open in the standard topology
@PeterTamaroff Take the space of all real valued functions on $[0, 1]$ with the topology of pointwise convergence (that is, the smallest topology that makes all of the functions continuous), this is not metrizable.
So, that means there is an $\epsilon_n > 0$ and a finite set $S_n$ such that $G_n \supset \{f : |f(x)| < \epsilon_n \text{ and } x \in S_n\}$.
For each $n$. So set $g(x) = 0$ for all $x$ in the $\cup S_n$ and $0$ otherwise. This will be in the intersection and is not $0$ hence not metrizable. Done.
@PeterTamaroff I hope I did not mess up. Anyway, it is called "topology of pointwise convergence" if you look it up you will probably find this too. It is also called "weak topology".
