Category Archives: Uncategorized

Every dcpo can be seen as a topological space, once we equip it with the Scott topology. And every topological space can be seen as a convergence space, so every dcpo can be seen as a convergence space.  In 2003, Reinhold … Continue reading →

Only a short post this month: I would like to explain Lawson’s construction of an FS-domain that is not known to be an RB-domain. Roughly speaking, this is the domain of closed discs of the under with reverse inclusion, and … Continue reading →

Characterizing properties of graphs, posets, and even dcpos by forbidden substructures is an intriguing approach.  Xiaodong Jia managed to show that every CCC of quasi-continuous domains must consist of continuous domains exclusively, and I would like to explain how this rests … Continue reading →

Meet-continuous dcpos were defined and studied by Hui Kou, Ying-Ming Liu, and Mao-Kang Luo about 14 years ago, and their importance only starts to be appreciated now.  One of the leading results in the theory of meet-continuous dcpos is that a dcpo is continuous if … Continue reading →

I have already mentioned Markowsky’s Theorem (1976): every chain-complete poset is a dcpo.  This is a non-trivial theorem, and I’ve given you a proof of it based on Iwamura’s Lemma and ordinals in a previous post.  Maurice Pouzet recently pointed … Continue reading →

At the start of the book, I had stated: “Topological convexity, topological measure theory, hyperspaces, and powerdomains will be treated in further volumes.”  The book got out in 2013, but I wrote that in 2011, almost seven years ago now.  What … Continue reading →

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 … Continue reading →