A hierarchy of clopen graphs on the Baire space
Arnold W. Miller

TL;DR
This paper investigates the structure of clopen graphs on the Baire space, establishing a hierarchy with uncountably many classes under continuous reduction and embedding, thus answering a question posed by Stefan Geschke.
Contribution
It introduces a hierarchy of clopen graphs on the Baire space and proves the existence of an uncountable family into which all such graphs embed, but not a countable family for reduction.
Findings
No countable family of clopen graphs covers all under continuous reduction.
Existence of uncountably many clopen graphs into which all others embed.
Answers a question of Stefan Geschke about the hierarchy of these graphs.
Abstract
We say that binary relation E on a space X is a clopen graph on X iff E is symmetric and irreflexive and clopen relative to X x X minus its diagonal. Equivalently for distinct x, y in X there are open sets U,V with (x,y) in U x V and either U x V a subset of E or U x V a subset of E complement. For clopen graphs E_1 and E_2 on the Baire space (omega^omega) we say that E_1 continuously reduces to E_2 iff there is a continuous map f from the Baire space to itself such that for [(x,y) in E_1 iff (f(x),f(y)) in E_2 ] for distinct x,y. Note that f need not be one-to-one but there should be no edges in the preimage of a point. If f is a homeomorphism to its image, then we say that E_1 continuously embeds into E_2. Theorem. There does not exist countably many clopen graphs on the Baire space such that every clopen graph on the Baire space continuously reduces to one of them. However…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Rings, Modules, and Algebras · Advanced Graph Theory Research
