Linear orders on chainable continua

Abstract

We define and study certain linear orders on chainable continua. Those orders depend on a sequence of chains obtained from definition of chainability and on a fixed non-principal ultrafilter on the set of natural numbers. An alternative method of defining linear orders on a chainable continuum $X$ uses representation of $X$ as an inverse sequence of arcs and fixed non-principal ultrafilter on $\mathbb{N}$. We compare those two approaches. We prove that there exist exactly $2$ distinct ultrafilter orders on any arc, exactly $4$ distinct ultrafilter orders on the Warsaw sine curve, and exactly $2^{\mathfrak{c}}$ distinct ultrafilter orders on the Knaster continuum. We study the order type of various chainable continua equipped with an ultrafilter order and prove that a chainable continuum $X$ is Suslinean if and only if for every ultrafilter order $\leq_{\mathcal{U}}^{\mathcal{D}}$ on $X$, the space $(X, \leq_{\mathcal{U}}^{\mathcal{D}})$ is order isomorphic to $([0,1],\leq)$. We study also descriptive complexity of ultrafilter orders on chainable continua. We prove that the existence of closed ultrafilter order characterizes the arc and we show that for Suslinean chainable continua, any ultrafilter order is both of type $F_{\sigma}$ and $G_{\delta}$. On the other hand, we prove that there is no analytic and no co-analytic ultrafilter order on the Knaster continuum.