The space S0

S₀ is a space that occurs in Matthew de Brecht’s generalized Hurewicz theorem for quasi-Polish spaces, published in 2018. S₀ is very simple: it is an infinite countably-branching tree, and if you order it so that the root is at the top, S₀ comes with the upper topology of the resulting ordering. S₀ is one of the four canonical examples of a non-quasi-Polish space (in a precise sense). I will describe it, and I will show how closed sets and compact saturated sets in S₀ can be described through certain kinds of subtrees. With that done, we will see that S₀ is sober, Choquet-complete, and completely Baire, but not locally compact, not convergence Choquet-complete, not compactly Choquet-complete, not LCS-complete, and, finally, not quasi-Polish. Read the full post.