Let $X=[0,1]^\omega$. We know that under the standard product topology, $X$ is metrizable. But is it metrizable under the box topology? My guess is not, and the intuition is that Urysohn's metrization theorem fails because the basis for $X$ under the box topology is uncountable. But that's not sufficient to disprove metrizability.
How can one show that this is indeed the case? (or prove otherwise)