A topological space is separable if it has a countable dense subset. A space is first countable if it has a countable basis at each point. It is second countable if there is a countable basis for the whole space. A collection of subsets of a space is locally finite if each point has a neighborhood which intersects only finitely many sets in the collection. A collection of subsets of a space is sigma-locally finite (AKA countably locally finite) if it is the union of countably many locally finite collections.
My question is, if a space is separable, first countable, and has a sigma-locally finite basis, must it also be second countable? I think the answer is yes, because I haven't found any counterexample here.
Any help would be greatly appreciated.
Thank You in Advance.
EDIT: I fixed my question. I meant that the space should have a locally finite basis, not be locally finite itself, which doesn't really mean much.