Order Topology on a Preorder

Question

While looking at the definition of the order topology defined on a total order (https://en.wikipedia.org/wiki/Order_topology), I realized I needed a generalization to preorders. So ultimately the question: Is there a generalization of the order topology to preorders. If so, what is it? If not, what are some workarounds?

Some motivation: My idea is to use “dense” preorders to model cause and effect where a<=b iff a causes b. The topology would make it possible to capture the idea that it is posible certian events are “closer” to directly causing another event.

Answer 1

There is the Alexandrov topology on a pre-ordered set.
It is not a generalization of the order topology you link to, but it might be what you're looking for

Answer 2

Define for each $x \in X$, the lower and upper sets $L(x) = \{y \in X: y < x\}$ and $U(x) = \{y \in X: y > x\}$. By definition all such sets together form a subbase for the order topology when $(X,<)$ is a linear order.

So if you want to generalise to a partial order $(X,\le)$, just define $x < y$ as $ x \le y$ and $x \neq y$ and use the same subbase.