I said earlier that what convergence was the starting point of topology. Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence directly, instead of through opens?
One bonus is that we shall obtain Cartesian-closed categories, and this will be easy! As usual, while nets will allow us to grasp the concept, this will actually work best with filters. See the full post.
It’s out! And freaky difficult as well… Play it online, too, and don’t forget the other topology-related crossword puzzles. Enjoy!
It’s been a long time, and I haven’t given any news from the Summer Topology Conference. There I met Frédéric Mynard. Frédéric stressed the importance of filters to me, and I should mention a few of the nice things one can do with them. See the first part of the post.
I’m off to the Summer Topology Conference, North Bay, Ontario, Canada. I’ll be giving two talks there. One is directly related to Chapter 7 of the book, and may help get a global vision of what’s going on there: have a look at the slides. I’m also going to give another nifty talk, on a short proof of the Schröder-Simpson theorem (see the slides to see what that is); but that is not related to the book, rather to its sequel. After that, I’m on holidays: happy holidays to you all as well!
Dito Pataraia once came up with an elegant proof of (essentially) the Bourbaki-Witt theorem. The proof is very short, and deserves a look. Furthermore, it uses domain theory in an essential way: the core is showing that the set of inflationary monotonic maps on a dcpo is itself a dcpo, and is directed, hence has a supremum. Dito Pataraia would very rarely publish his findings, and we owe it to Martín Escardó that it did not disappear with him. I personally became aware of this thanks to one anonymous referee of the book. See the full post.
The book is out! I’ve thanked several people at the beginning of the book, and I should now also thank everybody at Cambridge University Press, and notably Roger Astley, Helena Dowson, Clare Dennison, Caroline Mowatt, and Sarah Payne. It was a pleasure to interact with dedicated and serious people such as you!
There are now two extra menus here.
Let us skip the “course ideas” menu. You see what I mean. It might even be helpful.
Instead, look at the “puzzles menu” if you’d like some relaxation. All right, the intellectual kind: crossword puzzles! And related to the book‘s themes, at that. Enjoy!
A model of a space X is a dcpo Y whose subspace of maximal elements is isomorphic to X. Of particular importance are those spaces that have ω-continuous models. This topic is covered in Section 7.7.2 of the book. Martin [3], and Mummert and Stephan [1] came very close to characterize them exactly. See the full post.
Polish spaces are an important class of spaces. I am dealing with them in Section 7.7 of the book. But they are Hausdorff spaces, and the purpose of the book was to explore non-Hausdorff spaces. Matthew de Brecht found a non-Hausdorff generalization of Polish spaces, which he aptly called quasi-Polish spaces. See the full post.
This is all about “Non-Hausdorff Topology and Domain Theory – Selected Topics in Point-Set Topology“, a forthcoming book, to be published by Cambridge University Press.
See the table of contents.
