A hierarchy of clopen graphs on the Baire space

2012/10/31 by Arnold W. Miller, Miller, Arnold W.
Computer Science · Mathematics · #03E15 #Advanced Graph Theory Research #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1210.8362

openalex publication_date 2012/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 E₁ and E₂ on the Baire space (omega^omega) we say that E₁ continuously reduces to E₂ iff there is a continuous map f from the Baire space to itself such that for [(x,y) in E₁ iff (f(x),f(y)) in E₂ ] 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 E₁ continuously embeds into E₂. 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 there does exists omega₁ clopen graphs on such that every clopen graph continuously embedds into one of them. This answers a question of Stefan Geschke. Latex2e: 9 pages Latest version at: www.math.wisc.edu/~miller

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