A non metric first countable topological space

Question

Every metric space is first countable, but what about the converse? Does it always hold? If not, can anyone give a counterexample? Thanks

Answer 1

The long line is locally homeomorphic to $\mathbb R$ (and so first countable) but is not metrizable.

Less exotically, the lower limit topology on $\mathbb R$ is also first countable but not metrizable.

Answer 2

The converse is false. Take the Sierpinski space $X = \{0,1\}$ with topology given by $\tau = \{ \varnothing, \{0\}, \{0,1\} \}$. This topology is first-countable but it is not Hausdorff, hence not metrizable.