One thought I had this morning: the name “principal open set” seems a better (more descriptive) name than “distinguished open set”. I then googled the phrase to see if it had been used in some different way, and found to my surprise that it actually had been used in precisely in this way; I’m not sure where this usage originated.

But I’m inclined to switch over completely to this phrase. The clearest downside is that the notation D(f) refers to “Distinguished”. But I already prefer to think of it as the “Doesn’t-vanish” set.

This might suggest a better name for the following two “topologies”: the topology on {\rm {Spec}} A consisting of principal/distinguished open subsets; and the “topology” on a scheme X where the allowed open subsets are affine open subsets, and the allowed open morphisms are “principal/distinguished” inclusions. I’ve been calling the latter the “Distinguished Affine Topology”, and I’m not sure if I gave a name to the former. Are there better names for these?

Separately, now that AGITTOC (at least the first incarnation) is over, and I might write more here, as a world-readable (and world-commentable) notebook.

I’ve not posted a new version of the notes in a long time, because parts of it have been “closed for construction”. But I might resume soon. It might be easiest to post just a few chapters from the beginning, and gradually move forward.

As always, there are many comments here that I’ve not responded to. You might be surprised to find that I’ve actually read them, and made changes in response to many, and have intended changes in response to others.