Are requests of the form "has this ever been studied, and if so, may I please have a reference?" considered appropriate?
But let me be more specific.
So firstly, there's the point-set or "classical" approach to topology, which concerns itself with ordered pairs $(X,\tau)$ called topological spaces.
Then there's the pointless approach to topology, which concerns itself with lattices $(\tau,\wedge,\vee)$ called frames (in which finite meets distribute over arbitrary joins.)
I'm interested in a concept halfway between the two. We might call it "the classical approach, but with lattices." Rather than $(X,\tau)$, we concern ourselves $(P,\tau),$ where $P$ is a lattice that is isomorphic to a powerset lattice, and $\tau$ is a subset of $P$ that is closed with respect to arbitrary joins etc.
The motivation for this idea is as follows: we may be able to weaken the requirement that $P$ needs to be isomorphic to a powerset, and still be able to develop classical topology just fine.
So my question is, has this idea been studied before, and if so, may I please have reference recommended?