Is lower limit topology locally metrizable? I think that it is - for any $x\in \mathbb{R}$, $x\in [a,b)$ for some $a\in \mathbb{R}$ and $b\in \mathbb{R}$ , and $[a,b)$ as a subspace of $\mathbb{R}$ is metrizable. Is this proof correct?

  • $\begingroup$ Why bother with $[a,b)$? $\mathbb R$ is a neighborhood of $x,$ and $\mathbb R$ as a subspace of $\mathbb R$ is metrizable. $\endgroup$ – bof Feb 25 '17 at 4:27
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    $\begingroup$ @bof $\mathbb{R}$ with the lower-limit topology is not metrizable (it is separable but not second countable). As a matter of fact, I believe this shows it isn't locally metrizable either. Given $x\in \mathbb{R}$, it has a basic open set $[a,b)$ which contains it. But in the subspace topology, this remains to be separable but not second countable, so not metrizable. $\endgroup$ – Hayden Feb 25 '17 at 4:30

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