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Linear orders on chainable continua

Witold Marciszewski, Julia Ścisłowska, Benjamin Vejnar

TL;DR

The paper studies linear orders on chainable Continua induced by non-principal ultrafilters, formalizing two complementary constructions via chains and inverse limits. It computes precise counts of distinct ultrafilter orders for representative continua (2 on arcs, 4 on the Warsaw sine curve, and $2^{\mathfrak{c}}$ on the Knaster continuum) and establishes deep links between these orders and classical topological properties: Suslinian chainable continua yield order types isomorphic to $([0,1],\le)$ with a Borel (1,1) correspondence, while Suslinian-ness is equivalent to the order topology being CCC. The descriptive-set-theoretic analysis distinguishes arc-orders (closed $M$), Suslinian continua ($F_\sigma$ and $G_\delta$), and Knaster (non-analytic, non-co-analytic) cases, revealing a rich spectrum of complexity. The work also demonstrates that Knaster-order topologies are non-connected, non-compact, and non-separable, and proves a density-type transfer between inverse-limit ultrafilter orders and continuum ultrafilter orders, while posing numerous open questions about reversibility, density, and structure of ultrafilter orders on various 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 Suslinian if and only if for every ultrafilter order $\leq_{\mathcal{U}}^{\mathcal{D}}$ on $X$ the space $X$ with an order topology, generated by the order $\leq_{\mathcal{U}}^{\mathcal{D}}$, is ccc. 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 Suslinian chainable continua, any ultrafilter order is both of type $F_σ$ and $G_δ$. On the other hand, we prove that there is no analytic and no co-analytic ultrafilter order on the Knaster continuum.

Linear orders on chainable continua

TL;DR

The paper studies linear orders on chainable Continua induced by non-principal ultrafilters, formalizing two complementary constructions via chains and inverse limits. It computes precise counts of distinct ultrafilter orders for representative continua (2 on arcs, 4 on the Warsaw sine curve, and on the Knaster continuum) and establishes deep links between these orders and classical topological properties: Suslinian chainable continua yield order types isomorphic to with a Borel (1,1) correspondence, while Suslinian-ness is equivalent to the order topology being CCC. The descriptive-set-theoretic analysis distinguishes arc-orders (closed ), Suslinian continua ( and ), and Knaster (non-analytic, non-co-analytic) cases, revealing a rich spectrum of complexity. The work also demonstrates that Knaster-order topologies are non-connected, non-compact, and non-separable, and proves a density-type transfer between inverse-limit ultrafilter orders and continuum ultrafilter orders, while posing numerous open questions about reversibility, density, and structure of ultrafilter orders on various 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 uses representation of as an inverse sequence of arcs and fixed non-principal ultrafilter on . We compare those two approaches. We prove that there exist exactly distinct ultrafilter orders on any arc, exactly distinct ultrafilter orders on the Warsaw sine curve, and exactly 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 is Suslinian if and only if for every ultrafilter order on the space with an order topology, generated by the order , is ccc. 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 Suslinian chainable continua, any ultrafilter order is both of type and . On the other hand, we prove that there is no analytic and no co-analytic ultrafilter order on the Knaster continuum.
Paper Structure (16 sections, 28 theorems, 63 equations, 6 figures)

This paper contains 16 sections, 28 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The ultrafilter orders on chainable continua
  3. Structure of the paper
  4. Background on chainable continua
  5. Definitions of ultrafilter orders and some of their basic properties
  6. Examples of ultrafilter orders on simple chainable continua
  7. Ultrafilter orders on an arc
  8. Ultrafilter orders on continua $S_1,S_2$ and $S_3$
  9. Order type of ultrafilter orders on Suslinean chainable continua
  10. Ultrafilter orders on the Knaster continuum
  11. Order topology generated by a certain ultrafilter order on a Knaster continuum
  12. Descriptive complexity of ultrafilter orders on chainable continua
  13. Arc
  14. Suslinean continua
  15. The Knaster continuum
  16. ...and 1 more sections

Key Result

Theorem 3.4

A continuum $X$ is a chainable continuum if and only if it is homeomorphic to an inverse limit of sequence of arcs, i.e. a space $\varprojlim (X_i,f_i)_{i=1}^{\infty}$ for $X_i = [0,1]$.

Figures (6)

  • Figure 1: Example of chains $D = \{d_i\}_{i=1}^{k} \in \mathcal{D}$ and $E = \{e_i\}_{i=1}^{k}\in \mathcal{E}$.
  • Figure 2: Example of chains $D' = \{d'_i\}_{i=1}^{k} \in \mathcal{D}'$ and $E' = \{e'_i\}_{i=1}^{k}\in \mathcal{E}'$.
  • Figure 3: Chain $D_1$, covering interval $I_1$ according to $x_1 = 0$.
  • Figure 4: Chain $D_2$, covering intervals $I_1, I_2$ according to $x_1 = 0, x_2 = 1$.
  • Figure 5: Chain $D_3$, covering intervals $I_1, I_2, I_3$ according to $x_1 = 0, x_2 = 1, x_3 = 1$.
  • ...and 1 more figures

Theorems & Definitions (64)

  • Definition 3.1: Linear order on subsets of $X$
  • Definition 3.2
  • Theorem 3.4
  • Definition 3.5
  • Definition 3.6
  • Lemma 3.7: Lemma 2.4.20 in macias
  • Lemma 3.8
  • Theorem 3.9
  • proof
  • Theorem 4.1
  • ...and 54 more