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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 !

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    for anyone who's interested, meanwhile i found this, and Theorem 2.5 states necessary and sufficient conditions for a space to be second-countable : http://www.ams.org/journals/proc/1982-084-01/S0002-9939-1982-0633295-9/S0002-9939-1982-0633295-9.pdf – 2012-09-08
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    STILL, i would like to know, in the particular case when the space is Hausdorff, is there is any other condition wich is equivalent to be second-countable space. – 2012-09-08
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    The requirement in Theorem 2.5 that the family $\{B_p:p\in X\}$ be cohesive really just supplies the essential part of the metric space structure used in the proof that a separable metric space is second countable. It’s a case of *What do I need to make that proof work elsewhere?* – 2012-09-13

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