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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.

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No, these terms are not commonly used when $S$ is not assumed to be a topology.

Note that in the context of fields of sets (or more generally as long as $S$ is closed under complements), all three of these conditions are equivalent. A field of sets which satisfies these equivalent conditions is commonly said to separate points. More generally, "separating points" is sometimes used to refer to the $T_0$ condition for an arbitrary $S\subseteq P(X)$ (note though that this conflicts with the use of "separate" in topology where "separating points" would mean $T_2$).

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