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Compactness and Symmetric Well Orders

Abhijit Dasgupta

Abstract

We introduce and investigate a topological version of Stäckel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$ topological space $(X, τ)$ to be Stäckel-compact if there is some linear ordering $\prec$ on $X$ such that every non-empty $τ$-closed set contains a $\prec$-least and a $\prec$-greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor-Bendixson rank $< ω_2$ under ZFC. Under V=L, the equivalence holds in all scattered spaces.

Compactness and Symmetric Well Orders

Abstract

We introduce and investigate a topological version of Stäckel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a topological space to be Stäckel-compact if there is some linear ordering on such that every non-empty -closed set contains a -least and a -greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor-Bendixson rank under ZFC. Under V=L, the equivalence holds in all scattered spaces.
Paper Structure (7 sections, 20 theorems, 18 equations)

This paper contains 7 sections, 20 theorems, 18 equations.

Table of Contents

  1. Introduction and Summary of Results.
  2. Symmetric Well Orders and Stäckel-compactness.
  3. Basic Properties of Stäckel-compact spaces.
  4. The Case of Countably Compact Spaces.
  5. CB-rank $\omega_2$ and beyond.
  6. Stäckel-compactness and Novák Spaces
  7. Open Questions, Credits, and History

Key Result

Proposition 2.2

A set $X$ is finite if and only if a symmetric well-order can be defined on $X$.

Theorems & Definitions (41)

  • Definition 2.1
  • Proposition 2.2: Stäckel Stackel
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • ...and 31 more