Non-Hausdorff Topology and Domain Theory

Klaus Keimel passed away on Saturday, November 18th, 2017, and this is sad news.  I would like to pay homage to his memory, through a partial recollection of my own path with Klaus.

In the open problem section, I defined a FAC space as a topological space in which every closed subspace is a finite union of irreducible closed subspaces. FAC is for “finite antichain property”, since it generalizes the following theorem, due to Erdős and Tarski (1943): a poset has the finite antichain property (namely, all its antichains are finite) if and only if its downwards-closed subsets are finite unions of ideals. I asked about a similar characterization of FAC spaces. Let me give a positive answer to that in the full post!

I have just returned from the International Symposium on Domain Theory, which took place in Shijiazhuang, Hebei, China.  That was a fine conference indeed.  There, I met Xiaoyong Xi and Jimmie Lawson, who just happened to publish a remarkable result, related to a very recent post on coherence of dcpos: every complete lattice, and more generally every bounded-complete dcpo is well-filtered in its Scott topology.  Read the full post.

The nice thing about colleagues is that, sometimes, they give me a primer on their latest results.  I would like to talk about a strange result by Dongsheng Zhao and Xiaoyong Xi, which, while accepted for publication, does not seem to be out yet.  (Thanks to D. Zhao for letting me know about this!)  I have already talked about models of topological spaces.  Following earlier results by Zhao, Xi, and Erné, one can show that every T₁ space has a bounded complete, and even algebraic, poset model, and that every T₁ space has a (not bounded complete) dcpo model, but can we have both at the same time?  In other words, does every T₁ space have a bounded complete dcpo model?  Answer (and explanations) in the full post…

It had been a long time since I wanted to explain a nifty result by Jia, Jung, and Li (2016), which gives a simple test for whether a given well-filtered dcpo is coherent.  The proof, in particular, is extremely nifty.  Read the full post.

There are a few questions that I would like to solve, and I’ve decided to share them with you.  See the new open problems page.  I am starting with three questions now, but I will update that from time to time.  Some of these problems are pretty tough, and I’m asking for help here.  Some others are more doable, and indeed would be nice subjects for brilliant young M2 or PhD students.

— Jean Goubault-Larrecq

Happy New Year 2017!  Sorry I have not posted for some time…  Today, let me talk about a curious construction of Lawson and Xi on families of continuous maps that are not directed, rather pointwise directed.  They use that to show that the dcpo of all Scott-continuous maps from a core-compact, core-coherent space to an RB-domain is a continuous dcpo; but the idea of pointwise directed families of maps, and the way they use it to find a basis of dcpos of continuous maps, is intriguing.  Read the full post.

Last time, we had stated and proved the Dolecki-Greco-Lechicki theorem: every regular Čech-complete space is consonant.  I would like to show that there are some other classes of consonant spaces, among T₀ spaces.  The results are going to be easy consequences of results from the book.  Read the full post.