This question may be a little too easy for this site, but I'll ask it anyway: when is a Hausdorff topological space metrisable?
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The wikipedia page here is probably of interest. It contains links to a few metrization theorems, including results of NagataSmirnov's and Bing's. There is also the Smirnov metrization theorem that appears in Munkres' topology: it states that A space X is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable. 


I think it's also worth pointing out the Wikipedia page on Moore spaces. It turns out that the full answer to your question (if you don't allow conditions like "locally metrizable") has something to do with the Continuum Hypothesis. 

