Title: A hierarchy of clopen graphs on the Baire space
Author: Arnold W. Miller
Oct 2012
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
[(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 many clopen graphs 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