Ultrafilter order topologies on the pseudoarc

Describe the order topologies τ_U^D generated by ultrafilter orders ≤_U^D on the pseudoarc, including key properties such as connectedness, compactness, separability, character, and potential order-type or structural invariants.

Background

The pseudoarc is a classical example of an indecomposable chainable continuum with rich and subtle structure. The paper raises the problem of characterizing the order topologies produced by ultrafilter orders on the pseudoarc.

Resolving this would extend the descriptive and structural analysis of ultrafilter orders from arcs, Warsaw sine curve, and Knaster continuum to the pseudoarc, potentially revealing new phenomena specific to this continuum.