I call a family of subsets of a topological space separating if for any two distinct points there is a set in the family containing exactly one of them. I am looking for some (preferably equivalent) conditions under which a space admits a countable separating family of open sets.
If a space is $T_1$ and has a countable base then this countable base is such family. On the other side, any space admitting such family must be $T_0$.
Are there known any other necessary or suffcient conditions for the existence of such family? I am specially interested if the condition "separable and perfectly normal" is sufficient.
Note: If the space admits a countable family of open sets separating in a somewhat stronger sense "for any $x\neq y$ there is a set $U$ in the family such that $x\in U$ and $y\notin U$" then each singleton must be $G_\delta$ and hence the space must be $T_1$. As Henno Brandsma pointed out (his comment has disappeared), there is a cardinal characteristic called separating weight, defined as the minimum cardinality of a separating family of open sets, in this stronger sense. Actually, this stronger condition was what I had originally on mind when I asked this question.
I'll put my comments in an answer:
The minimal size of such a point separating family (also called a pseudobase) is called either $\psi w(X)$ (pseudoweight, Juhasz) or $sw(X)$ (separating weight, Hodel). So you ask for conditions that $\psi w(X) = \aleph_0$. $X$ must be $T_1$ for this to be defined at all, as you state.
For $T_2$ spaces we have $\psi w(X) \le nw(X)$ (Juhasz 2nd cardinal functions book 2.8.b), so a $T_2$ space with countable network (a collection like a base but they need not be open sets) will have such a countable family.
A necessary condition is also $|X| \le 2^{\aleph_0}$, or more accurately $\psi w (X) \le \log(|X|)$. A nice paper on this and related cardinal functions is from Brian M. Scott (also active here). He names this cardinal function $\psi_1(X)$ ($1$ refers to $T_1$).