Let $\left<X,S\right>:S \subseteq 2^X$. Can names relating to separation axioms be used to describe $S$ without further explanation, even if $S$ does not form a topology? For instance:
$S$ is $T_0$ or Kolmogorov - for any $x,y \in X: x \neq y$ there is $A \in S$ that contains only one of the two.
$S$ is $T_1$ or Frechet - for any $x,y \in X : x \neq y$ there is $A \in S$ that contains $x$ but not $y$.
$S$ is $T_2$ or Hausdorff - for any $x,y \in Y: x \neq y$ there are $A,B \in S$ such that $x\in A$, $y \in B$ and $A \cap B = \emptyset$.
Or are there, perhaps, some other names for such properties in more general, non-topological frame? I am particularly interested in fields of sets.