Is there any "second-countable" theorem ? With this i mean if there is any result like Nagata-Smirnov Theorem (that states necessary and sufficient condition for a space be metrizable), but for second-countable spaces. Also, with Urysohn Metrization Theorem it's straightforward to prove that if a space is compact and Hausdorff, then is secound countable iff is metrizable. Is there any result like this but with the hypothesis that the space is only Hausdorff (i mean, something like : Let X be a Hausdorff space. Then X is second-countable iff [something]) ?
Thanks a lot !