Let us say a topological space $X$ is a *countable union of second countable spaces* if there exists a sequence of subsets $\{X_n\}$ of $X$ with $X=\cup X_n$ such that  the relative topology on $X_n$'s is second countable. Clearly $X$ will be *separable*. 

Q. What  topological properties **P** are strictly  between separabilty and countable union of second countable spaces?

Countable union of $2^{ed}$-countable spaces $\prec$ **P** $\prec$ Separablity