I would like to be able to cite an introduction to continuous lattices that is written for (and ideally by) real analysts. So far, my enquiries amongst the experts on continuous lattices have drawn a blank, but maybe some analyst has had occasion to use them, or maybe teach a graduate course about them.
Define "real analyst." These range from the practical cowboys to the sensitive constructivists (though as Hollywood reminds us the intersection need not be empty). This distinction persists in computational analysis, with Blum-Shub-Smale representing the cowboys and Metropolis, Rota, Edalat, Escardo, Freyd, Leinster, etc. bringing up to date the descriptive set theory program started by Borel, Baire, and Lebesgue. The Compendium came out in 1980. Maybe to those of you on the right hand side of the Atlantic it might have seemed to be addressing computer scientists, but to most of us in the Western hemisphere (pace Wand, Tennent, and a couple of others) it looked like it was written for analysts. I doubt if you're going to find a treatment written *more* for analysts than the Compendium and its updates and successors. For your purposes its three downsides might be its length, its datedness (not so dated remarkably when you consider how new the subject was then and how much has been learnt since), and its relative inaccessibility (~$100 for second-hand copies in good condition, $50 for a solitary "acceptable" copy, ~$160 for the new books). The Wikipedia article on Lattices (order) has a brief introduction to continuous and algebraic lattices that might hold the fort---if two more sentences would do the trick add them yourself, no one will stop you. Then there's the longer article on Domain Theory. It's hard to imagine any analyst who's likely to be interested in abstract domain theory not being willing to tackle the domain theory article on its own merits, recognizing the intrinsically computational aspects of constructive analysis, at least as the computing professionals see it. The modern constructive analyst is going to have to merge the paradigms of analysis and computation in order to keep up with where computer scientists have been pushing the subject. Don't pander to the retards. Vaughan