The topological definition of convexity is
Let $Y \subset X$ a convex set. If $a,b \in Y$ and $a < c <b$, then $c \in Y$.
For that, we need the order topology.
Is it possible to define convexity for any topological spaces? Or for more spaces than just those with the order topology.
This definition being equivalent to the definition I mentioned when space is well ordered.
There is a general topology flavoured definition of a general convexity space, which generalises the already existing notions of convexity that have been defined in the past.
The intuitive definition: "if two points $x,y$ are in a convex set, then all points "in between" $x$ and $y$ must also be in that set. Such sets are called convex sets.
This is clearly true for $\emptyset$ (voidly) and $X$, regardless how we would define "in between". Convex sets are also quite clearly (from the logical form of the definition) closed under arbitrary intersections (like closed sets in a topological space are) and this allows us to define a "convex closure" (usual called "convex hull") for any subset, e.g. The final axiom, which also follows from the logical structure of its definition, is that a union of any chain of convex sets (a chain means that for any two sets in the chain, one of them is contained in the other) is also convex. This is in stark contrast with closed sets in a topology, where this certainly does not hold, unless in simple cases like the discrete or indiscrete topology, e.g.
The spirit of this abstract definition is quite "topological". There are quite a few ways that have been studied to define this "in between" idea:
the classic one: in a vector space over $\Bbb R$ we define all points $tx + (1-t)y$ for $t \in [0,1]$ to be in between $x$ and $y$ and we get the classic notion of convexity (in the plane or 3-space, where it started, very geometrical).
Your own example: in a partial or linear order we can define $[x,y] = \{x \mid x \le z \le y\}$ to be the set of in between points. This gives order convexity.
In a metric space $(X,d)$ we can define a "metric segment" $[x,y]_d = \{z \in X \mid d(x,z) + d(z,y) = d(x,y)\}$ as the points in between $x$ and $y$ (based on both the intuition that the "degenerate triangles", where the triangle inequality is exactly an equality, are the equivalent of a straight line in that metric, as oging via such a $z$ is not a "detour"; and the fact that in the Euclidean metric this si jsut the first example all over again. This is the "geodesic convexity" induced by $d$. This can be applied to graphs in their natural distance, e.g.
A median operator also induces a convexity, but a "ternary" one: $C$ is convex for it, iff for all $x,y,z \in C$ we have that $\langle x,y,z\rangle \in C$ too.
Etc, etc. For many more examples and lots of theory see the book
M. van de Vel, Theory of Convex Structures. (written by an old teacher of mine from uni)
which has whole chapters on what happens with having a topology and a convexity on the same set and how they interact. (is the closure of a convex set again convex? Is the convex hull a continuous operation (in the hyperspace, eg.) etc. that kind of thing.
You cannot always define a "compatible" (depending on your definition) convex structure on a topological space I think, but the book does have a chapter on the "intrinsic topology" of a convex space. Check it out..