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In Munkres, section 30, exercise 6 is this:

Show that $R_{l}$ and $I^2_0$ are not metrizable.

I guess $R_{l}$ is lower limit topology, and $I^2_0$ is an ordered square. and here, how to prove they are not metrizable?

I read section 30 and 21 which is about the metric topology, got no clue.

According to the definition of "metrizable", I have to show that they have no metric, but then, I don't know why the usual metric(absolute value) is not a metric for them.

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    $\begingroup$ The usual metric would generate the usual topology...... $\endgroup$ – user99914 Nov 23 '14 at 13:23
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Show that $$(\mbox{separable $+$ metrizable} ) \Rightarrow \mbox{second countable}$$ and show that $\mathbb{R}_l$ is separable ($\mathbb{Q}$ is dense) and not second countable. Conclude that $\mathbb{R}_l$ is not metrizable.

Secondly, show that $I^2_0$ is compact, but it has an infinite sequence with no convergent subsequence. This means that $I^2_0$ is not metrizable.

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