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

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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.

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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.

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