Locally metrizable but not metrizable space I am looking for spaces which is locally metrizable without being metrizable. Here are the definitions:

Definition. A topological space $(X,\tau)$ is called metrizable if we can define a metric on $X$ that generates $\tau$. A topological space is called locally metrizable if every point has a metrizable neighborhood.

I actually know an example, namely the long line. In this case I would say the reason is that the long line is (in some sense) too long to meaningfully define distancies between its points with just using the real numbers.
So I am actually looking for examples of another kind (whatever this means). I know, this is not very rigorous, but maybe someone already knows significantly different examples.

Question: Are there locally metrizable but not metrizable spaces that are not "too big" for a metric but fail to have one for another reason? If not, can this be made rigorous in some way?

If something like this is possible, I am looking for a list of "reasons" why a space can fail to be metrizable while still being locally metrizable. Being "too big" might be one of them.

The line with two origins is locally metrizable but not even Hausdorff. Here the problem is local, in contrast to the long line.
