# Can separation axioms be used beyond topology without explanation?

Let $$\left:S \subseteq 2^X$$. 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.

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 $$T_2$$).