The following tiny mini-theory of convexity, which I've introduced in 1961, may serve as a start-point. It is abstract. However you may add an algebraic structure, topology, and whatever you wish, together with proper assumptions (e.g. local convexity in linear spaces) to get results in the respective environment.
Consider a pair $\ (X\ \mathcal C)\ $ consisting of a set $X$ and a family $\mathcal C$ of subsets of $X$. Call these subsets convex.
Definition 1 A set $\ H\subseteq X\ $ is called a half-space
$\ \ \Leftarrow:\Rightarrow
\ \ H\in\mathcal C\,\ \mbox{and}\,\ X\setminus H\in\mathcal C$.
Definition 2 Let $\ \emptyset\ne A\subseteq X,\ $ and
$\ a\in X\setminus A.\ $ Then the shadow $\ S(a\ A)\ $ of $\ A\ $
from $\ a\ $ is defined as follows:
$$ S(a\ A)\,\ :=\,\ \left\{x\in X:\ \forall_{D\in\mathcal C}\ \left(
\{a\,\ x\}\subseteq D\ \Rightarrow\ A\cap D\ne\emptyset\right)\right\} $$
There are $4$ axioms:
- $ \forall_{x\in X}\,\ \{x\}\in\mathcal C $
- $ \forall_{\mathcal D\subseteq\mathcal C}
\,\ \bigcap \mathcal D\in\mathcal C\,\ $ (hence $\ X\in\mathcal C$)
- $ \forall_{A\subseteq X}\ ((\forall_{x\ y\in A}
\exists_{D\in\mathcal C}\ \{x\ y\}\subseteq D)\ \Rightarrow
\ A\in\mathcal C) $
- $ \forall_{A\in\mathcal C\setminus\{\emptyset\}}
\forall_{a\in X\setminus A}\,\ S(a\ A)\in\mathcal C $
THEOREM
$$ \forall_{A\ B\in\mathcal C\setminus\{\emptyset\}}
\,\ ((A\cap B=\emptyset)
\ \ \Rightarrow\ \ \exists_{G\ H\in\mathcal C}
\ (A\subseteq G\ \mbox{and}\ B\subseteq H\ \mbox{and}\ G\cap H=\emptyset)
\,) $$
Remark 1
The convex sets $\ G\ $ and $\ H\ $ from the theorem are
half-spaces which separate disjoint convex sets $\ A\ $ and $\ B,\ $
i.e. every disjoint pair of non-empty convex sets can be separated by half-spaces.
Remark 2 The above axioms admit an equivalent elementary version which does not use the notion of sets but only a kind of an in betweenness relation.
