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Flimsy Spaces

Robin Khanfir, Béranger Seguin

TL;DR

The paper investigates n-flimsy spaces, proving there are no $n$-flimsy spaces for $n\ge 3$, and provides a complete classification of compact $2$-flimsy spaces as dense, order-complete cyclically ordered sets with their order topology. It develops an axiomatic framework of connectivity spaces to unify traditional connectedness, big-path-connectedness, and path-connectedness, and links these notions to order-theoretic structures via dense, order-complete separation relations and big circles. A central achievement is the triple equivalence among big circles, separation relations, and 2-flimsy connectivity spaces, yielding circle-like classifications and a clear correspondence between topological and order-theoretic circle models. The paper also constructs a 2-path-flimsy space that is not 2-big-path-flimsy, illustrating the independence of the three notions and enriching the landscape of circle-like topologies with intricate path phenomena. Overall, these results illuminate how circle-like behavior emerges from minimal flimsiness axioms and how order-theoretic and topological perspectives coherently describe this phenomenon.

Abstract

We study $n$-flimsy spaces, which are the topological spaces that remain connected when removing fewer than $n$ points but become disconnected when removing exactly $n$ points. We show that no such space exists for $n \geq 3$, and that the compact $2$-flimsy spaces are precisely the dense and order-complete cyclically ordered sets equipped with their order topology. Furthermore, we examine variants of the definition obtained by replacing connectedness by path-connectedness, where paths are either parametrized by $[0,1]$ or by arbitrary compact linear continua.

Flimsy Spaces

TL;DR

The paper investigates n-flimsy spaces, proving there are no -flimsy spaces for , and provides a complete classification of compact -flimsy spaces as dense, order-complete cyclically ordered sets with their order topology. It develops an axiomatic framework of connectivity spaces to unify traditional connectedness, big-path-connectedness, and path-connectedness, and links these notions to order-theoretic structures via dense, order-complete separation relations and big circles. A central achievement is the triple equivalence among big circles, separation relations, and 2-flimsy connectivity spaces, yielding circle-like classifications and a clear correspondence between topological and order-theoretic circle models. The paper also constructs a 2-path-flimsy space that is not 2-big-path-flimsy, illustrating the independence of the three notions and enriching the landscape of circle-like topologies with intricate path phenomena. Overall, these results illuminate how circle-like behavior emerges from minimal flimsiness axioms and how order-theoretic and topological perspectives coherently describe this phenomenon.

Abstract

We study -flimsy spaces, which are the topological spaces that remain connected when removing fewer than points but become disconnected when removing exactly points. We show that no such space exists for , and that the compact -flimsy spaces are precisely the dense and order-complete cyclically ordered sets equipped with their order topology. Furthermore, we examine variants of the definition obtained by replacing connectedness by path-connectedness, where paths are either parametrized by or by arbitrary compact linear continua.

Paper Structure

This paper contains 35 sections, 56 theorems, 41 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Other notions of connectedness.
  4. Main results
  5. Connectivity spaces, informally
  6. Illustration of the axioms.
  7. Comparison of connectivity spaces with other notions.
  8. Outline
  9. Acknowledgements
  10. Preliminaries
  11. Linear continua, big intervals, big paths
  12. Big circles
  13. Cyclic orders, separation relations
  14. Cyclic orders.
  15. Separation relations.
  16. ...and 20 more sections

Key Result

Theorem 1.3

Let $n \geq 3$ be an integer. There are no $n$-flimsy spaces, no $n$-big-path-flimsy spaces, and no $n$-path-flimsy spaces.

Figures (7)

  • Figure 1: In this example, removing $x$ yields three components $C_1$, $C_2$ and $C_3$, and indeed removing $C_i$ (for any $i \in \{1,2,3\}$) does not disconnect the figure.
  • Figure 2: A "button" situation.
  • Figure 3: A "seam" situation. The red (dotted) region $A$ consists of the bottom horizontal edge together with the red vertical stripes, and the blue (non-dotted) region $B$ consists of the top edge together with the blue vertical stripes. If the red and blue stripes alternate "densely often" and we are thinking in terms of (big-)path-connectedness, then it is impossible to find a button as in \ref{['fig-button']}, but a seam $S$ does indeed exist.
  • Figure 4: The three possible meanings of $[x,y,z]$ when $x,y,z \in X\setminus\{\mathrm{e}\}$.
  • Figure 5: A sketch of the situation of the proof of \ref{['only-two-components']}.
  • ...and 2 more figures

Theorems & Definitions (127)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: cf. \ref{['thm:no-3f-cspace']} and \ref{['prop:top-is-cspace', 'prop:path-is-cspace', 'prop:continuum-is-cspace']}
  • Theorem 1.4: proved in \ref{['sn:2f-top']}
  • Theorem 1.5: proved in \ref{['sn:2f-big-path']}
  • Theorem 1.6: proved in \ref{['sn:2f-path']}
  • Theorem 1.7: proved in \ref{['sn:equi']}
  • Theorem 1.8: cf. \ref{['thm:2pf-is-compact']}
  • Definition 2.1
  • Definition 2.2
  • ...and 117 more