Countable character of the ultrafilter order topology

Determine whether, for every chainable continuum X and every ultrafilter order ≤_U^D on X, the order topology τ_U^D has countable character at every point; that is, whether each point in (X, τ_U^D) admits a countable local base.

Background

Understanding local base properties (character) of the order topology induced by ultrafilter orders is fundamental for assessing metrizability and other topological features. The authors pose whether countable character holds universally in this setting.

Answering this would clarify how the ultrafilter order topology compares to classical order topologies on separable continua and could impact descriptive set-theoretic analyses presented earlier in the paper.