I am working in the context of binary rooted trees and I think I am probably reinventing the wheel with a large part of what I am doing.
More precisely, consider a full binary rooted tree $(T,<)$ (where $<$ is the tree order) such that every chain in $(T,<)$ has a supremum. (for instance $2^{\leq\omega}$ where $u<v$ iff $u$ is an initial segment of $u$)
I am looking at the topology on $(T,<)$ whose closed sets are those which are stable under suprema of non-empty chains.
Where can I find literature about such structures?