16:00
It is quite easy to see that the sobrification of a
topological space is a dcpo with respect to its specialisation order
and that the topology is contained in the Scott topology wrt this
order. It is also known that many classes of dcpo's are sober when
considered as topological spaces via their Scott topology. In 1982,
Peter Johnstone showed that, however, not every dcpo has this
property in a delightful short note entitled "Scott is not always
sober".
Weng Kin Ho and Dongsheng Zhao observed in the early 2000s that the
Scott topology of the sobrification of a dcpo is typically different
from the Scott topology of the original dcpo, and they wondered
whether there is a way to recover the original dcpo from its
sobrification. They showed that for large classes of dcpos this is
possible but were not able to establish it for all of them. The
question became known as the Ho-Zhao Problem. In a recent
collaboration, Ho, Xiaoyong Xi, and I were able to construct a
counterexample.
In this talk I want to present the positive results that we have about
the Ho-Zhao problem as well as our counterexample.