As I understand it, there are three canonical textbooks on pointless topology: the classic "Stone Spaces" by Johnstone, "Topology via Logic" by Steve Vickers, and the newer "Frames and Locales" by Picado and Pultr. I am curious for a comparison between the emphases of these three books. As far as prerequisites go, I have a reasonable background in category theory, though no exposure to sheaf theory, and have seen enough point-set topology to understand Stone-Čech compactification and Tychonoff's theorem. I have also seen a bit of commutative algebra and am willing to do additional reading to catch up on anything necessary. I have no background in lattice theory, but I think all three texts have either appendices or introductory chapters on the subject.
Having glanced through their indices, I have a vague impression of the material covered in each text. For instance, Johnstone's book contains a great deal of material on representation theorems, and also a chapter devoted to applications of locale theory to representations of rings (including discussion on Zariski and Pierce spectra); as far as I can tell this is absent from "Frames and Locales". "Topology via Logic" also has some discussion of Zariski spectra as well as a number of applications to theoretical computer science. On the other hand, "Frames and Locales" gives discussion on constructive mathematics (in particular why locale theory is a nice language for topology in constructive mathematics), while (again as far as I can tell) this is missing from the other two books.
In short, my question is: for which reader is each text most suitable, and which text would be the best self-contained introduction to the field for someone with my background. Apologies if this is not appropriate for the site, but I've struggled to find any extensive discussion comparing the three books.
Best Answer
Topology via Logic - theoretical computer scientist
Stone Spaces - pure mathematician
Both are really good. Topology via logic as it gives a good account of domain theory, including power domains.
The first few chapters of Stone Spaces really have no equal. Comprehensive, starting at the basics, excellent narrative flow and take you to the heights (I would even recommend it as an introduction to the basic constructions of sheaf theory).
Enjoy!