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I will denote the real numbers with the $K$-Topology as $R_{K}$ (If someone doesn't know or remember this topology, read here). I understand that $R_{K}$ is not regular, since the set $K$ cannot be separated from the point $0$, and it is second-countable, since you can take as basis the intervals with rational endpoints. It seems to me that this is not enough to prove that $R_{K}$ is not metrizable, but I don't know how to proceed. Someone suggested me to use Urysohn metrization theorem, but I think it is not possible in this case since it is not a characterization of metric spaces.

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Every metric space is normal, and hence regular. But the $K$-topology is not regular, thus not metrizable.

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