Probably the ideal book would have Stone duality as an advanced chapter? But I don’t really know enough to know what I want. I suspect I want the least advanced book among those that (1) defines a distributive lattice, and (2) has the word “continuous” somewhere in it.
The 1989 book "Topology via Logic" by Steven Vickers might be a good fit. Or a horribly bad one.
The set of textbooks not being a lattice, it could be that so such textbook exists.
My highly nonexpert impression is that interest in lattices declined in the second half of C20, and that they do not interact particularly well with topology. So it may be too much to expect a contemporary textbook covering both. Nachbin *Topology and Order* might be worth looking at, but it's old.
Priestley & Davis, *Intro to Lattices and Order* is the classic intro to lattices, but one of those intro books that is also useful for researchers, so it's not such easy reading. It contains some topology, but mostly in specific applications.
There's of course a ton of work in econ, with good survey articles, on when some kind of continuity condition on a partial order ensures some kind of representability. Aliprantis & Tourky *Cones and Duality* discuss partial orders generally, lattices in particular, and topology a lot.
Quite readable, but restricted to vector spaces, so the partial orderings are assumed to be linear.
Finally, haven't read it, but Johnstone, *Stone Spaces* might be perfect if it didn't have the reputation of being very difficult. Great outline of the history of the subject though.
Thank you! This is all super helpful.