Characterizing non-ω-well-filtered spaces by forbidden subspaces

In a 2023 paper, Hualin Miao, Xiaodong Jia, Chong Shen and Qingguo Li characterized ω-well-filtered spaces as those that do not contain any one of two very simple spaces, S₁ and S_D, as Skula-closed subspaces. This is part of a family of theorems characterizing classes of spaces through forbidden subspaces, and is, I think, a good first step in explaining Matthew de Brecht’s characterization of quasi-Polish spaces through forbidden subspaces, something I aim to be able to do someday. I will also take the opportunity to give a characterization of those first-countable spaces that are sober, due to Xiaoquan Xu, Chong Shen, Xiaoyong Xi, and Dongsheng Zhao in 2020. Read the full post.