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.


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 $T_2$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.