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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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    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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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)?

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    It would have been nice if there something simpler, but as there are countable non-second countable spaces (that are also normal etc.) no other countability property would do (like ccc, separable, countable network etc.) For compact Hausdorff spaces a countable network (like a base, but the sets need not be open themselves) does imply a countable base, but not in general.2012-09-10