Classification Problems in Continuum Theory

We study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics. We provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes. For dendrites with finitely many branch points the classes of homeomorphism and quasi-homeomorphism coincide and the minimal quasi-homeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the relation of homeomorphism between dendrites is S∞-universal. It is shown that the classes of trees and graphs are both D2(Σ 0 3)-complete, the class of dendrites is Π 0 3-complete, the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is Π3-complete. We also show that if G is a nondegenerate finitely triangulable continuum, then the class of G-like continua is Π2-complete.