Definition of topology using separation as primitive notion In "General Topology", chapter 1, exercise b, Kelley wrote in a note that it is possible to use the notion of separation ($A$ and $B$ are separated iff $A^k\cap B=A\cap B^k=\emptyset$) as primitive to define topological spaces and he put these three bibliographical references:
Wallace: "Separation Spaces"
Krishna Murti: "A set of axioms for topological algebra"
Szyrmanski: "La notion des ensembles separé comme terme primitif de la topologie"
I read the Wallace's article, but his definitions of separation as primitive only work for defining $T_1$-spaces (i.e., in that singletons are closed).
Moreover, I was not able to find on internet the other two articles.
How can I adjust the Wallace's definition so that it works for defining any topological space?
$\textbf{Note:}$
Here is Wallace's definition:
Given a set $X$, a separation relation is a relation $s\subseteq\mathcal{P}(X)\times\mathcal{P}(X)$ such that:
1) $\emptyset\,s\,A$.
2) If $A\,s\,B$, then $B\,s\,A$.
3) If $A\,s\,B$, then $A\cap B=\emptyset$.
4) If $A\,s\,B$ and $C\subseteq A$, then $C\,s\,B$.
5) If $A\,s\,C$ and $B\,s\,C$, then $A\cup B\,s\,C$.
6) If $\{x\}\,s\,A$, then $\{x\}\,s\,\{y\in X:\neg\{y\}\,s\,A\}$.
7) If $x\neq y$, then $\{x\}\,s\,\{y\}$.
8) If for all $x\in A$ and all $y\in B$ we have $\{x\}\,s\,B$ and $\{y\}\,s\,A$, then $A\,s\,B$.
Then taking $A^k=\{x\in X:\neg\{x\}\,s\,A\}$, it would be a closure operator, but the resulting topological space would be $T_1$.
 A: It can't be done.  For instance, let $X=\{0,1\}$ consider the topologies $\tau_0=\{X,\{0\},\emptyset\}$ and $\tau_1=\{X,\emptyset\}$ on $X$.  Note that $\tau_0$ and $\tau_1$ have exactly the same separated sets: namely, $A,B\subseteq X$ are separated iff at least one of them is empty.  So for non-$T_1$ spaces, a topology cannot always be recovered from its separation relation.
A: Note that there exists also a dual notion of proximity: basically $\def\p{\mathbin{p}}\def\s{\mathbin{s}}\def\clo{\overline}\def\set#1{\{#1\}}A\p B \iff ¬A\s B$, and its axioms are just translations of the axioms you gave (and by excluding or including some axioms in the list you get what is called Lodato proximity or Efremovič proximity).
The desired relation between the topology and the separation is $x ∈ \clo{A} \iff ¬\{x\}\s A \iff \{x\}\p A$.
As was observed the given axioms always give a $T_1$ space. Namely, the axiom 7 gives us $T_1$ and the axiom 2 gives us symmery. Note that if 7 was formulated $x ≠ y \implies (\set{x}\s\set{y}) ∨ (\set{y}\s\set{x})$, then it would imply just $T_0$, but in combination with 2 it still gives $T_1$.
So if you want to be able to obtain every topology, drop 2 and 7. On the other hand, if you put $A \s B \iff A ∩ \clo{B} = ∅$, you'll reconstruct the original topology. But it would be needed to investigate, which axioms the separation satisfies. Also, when you drop symmetry, you'll need to be more careful in formulating the other axioms.
In situation like this I like the systematic approach that you don't start with the list of the axioms, but rather with the construction that assign the candidate for a closure operator to every binary relation on $\mathcal{P}(X)^2$, and then you prove propositions like “if the separation satisfies this, then the operator satisfies that” and “the operator satisfies this if and only if the separation satisfies that”. Then you know which properties you need to obtain a topology and which properies are extras giving you $T_0$, symmetry or normality.
