I'm looking for a reference, if such a references exists.
So there are currently at least two approaches to topology.
The point-set or "classical" approach to topology, which concerns itself with ordered pairs $(X,\tau)$ called topological spaces.
The "pointless" approach to topology, which concerns itself with (particular kinds of) lattices $(\tau,\wedge,\vee)$ called frames. (For more information, see e.g. Wikipedia.)
I'm interested in a concept halfway between 1 and 2. We might call it "the classical approach, but with lattices."
In particular, rather than studying point-set topological spaces $(X,\tau)$, we concern ourselves with "lattice-theoretic" topological spaces $(P,\tau)$, 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 main motivation: We may be able to weaken the requirement that $P$ needs to be isomorphic to a powerset, and thereby obtain a more general theory, which is still classical in flavor.
Has this idea been studied before? If so, a reference recommendation would be great.