A topological space is second countable if it has a countable basis. A space is seperable if it has a countable dense subset. Now being second countable implies being seperable, but the converse doesn't hold in general. My question is, what is the weakest condition you need to add to seperability in order to imply second countability? It's not enough for a space to be first countable, because ℝ with the lower limit topology is first countable and seperable but not second countable. On the other hand, a seperable metric space is always second countable. So is there a condition, stronger than being first countable but weaker than being metrizable, which is sufficient to ensure that being seperable implies being second countable? Wow, that was a tongue twister! Any help would be greatly appreciated. Thank You in Advance.