Locally metrizable but not metrizable space

Question

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.

Answer 1

A nice overview paper by David Gauld, whose specialisation is exactly this, gives a lot of equivalent conditions for a manifold (locally Euclidean, connected Hausdorff space for him) to be metrisable.

See here and his book Non-metrisable manifolds, as well. Lots of references there.

A classical metrisation theorem (Smirnov): a locally metrisable Hausdorff space is metrisable iff it is paracompact.

Answer 2

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