A hierarchy of clopen graphs on the Baire space

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 there does exists omega_1 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