TITLE: About Countable Basis
QUESTION [8 upvotes]: 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 !

REPLY [3 votes]: What would be the use? Being second countable is hardly a difficult thing to check: find a countable base. The nice thing about the Bing-Nagata-Smirnov theorem is the fact that it replaces the "hard" task of finding a metric that is compatible with the topology by the "easier" task of finding a specific type of base (plus checking $T_3$) (This is only "easier" depending on how the space is defined in the first place, of course; but this theorem is also used to prove many other metrization theorems, and as such is quite useful). Moreover the metric is something external to the space (for one thing, it needs the reals as the codomain) while the base is internal to the space. The same holds for the special case of Urysohn: we have a countable base (easy to check) + $T_3$ (ditto) and get "for free" a compatible metric on the space. 
So my question to you is what would be a simpler condition that second countable (the coherence property you linked to seems to be less simple)?