First-countable spaces and their Smyth powerdomain

This month, we will look at certain conditions recently found by He, Li, Xi and Zhao in 2019, and then by Xu and Yang in 2021, in order to ensure that the Smyth powerdomain Q(X) (with the Scott topology) of a space X is a dcpo model of X, and in order to ensure that the Scott and upper Vietoris topologies agree on Q(X)… without the usual condition of local compactness, but under some first-countability conditions. I got interested in this while Xiaodong Jia and I explored the Smyth powerdomain of the Sorgenfrey line. Read the full post.