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.

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