My talk will be devoted to discuss families of ultrafilter orders on a given chainable continuum X (such as e.g. arc, the Warsaw sine curve, the Knaster continuum etc.). These orders depend on a fixed sequence of chains, covering X (obtained from chainability of X), and on a fixed nonprincipal ultrafilter on N. Alternatively ultrafilter orders may be defined using representation of X as an inverse limit of a sequence of arcs and a fixed nonprincipal ultrafilter on N.

During the talk I will present some known results in this topic. In particular, I will mention some ideas how we can express the “level of complexity” of a given chainable continuum in the language of ultrafilter orders.

This is a joint work with Witold Marciszewski and Benjamin Vejnar, preprint is available at: https://arxiv.org/abs/2510.14577.