Alberto Marcone

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings ≼∃∀ and ≼∀∃ on the power set of a quasi-ordering (Q,≼). We set X≼∃∀Y if every x∈X is majorized in ≼ by some y∈Y, and X≼∀∃Y if every y∈Y is minorized in ≼ by some x∈X. We show that both these quasi-orderings are α-wqo if and only if the original quasi-ordering is (α ⋅ ω)-wqo. For ≼∀∃ this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.

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 homeomorphism and quasi-homeomorphism classes coincide, and the minimal quasi-homeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the homeomorphism relation between dendrites is

Invariantly universal analytic quasi-orders

S_{\infty }

Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic

-universal. It is shown that the classes of trees and graphs are both

The maximal linear extension theorem in second order arithmetic

D_2( \boldsymbol \Sigma^0_3)

Interval Orders and Reverse Mathematics

-complete, the class of dendrites is

Linear extensions of partial orders and reverse mathematics

\boldsymbol \Pi^0_3

Extensions of functions which preserve the continuity on the original domain

-complete, and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is

Complexity of Sets and Binary Relations in Continuum Theory: A Survey

\boldsymbol \Pi^0_3

Epimorphisms Between Linear Orders

-complete. We also show that if